First I would say that during the medieval period the demand for books was much lesser than after the books became comparatively cheap after Gutenberg. The key here is the expense a quickly worn out woodblock presents that could only be used for one page. Whereas you already mentioned the crucial innovation for European book printing Gutenberg is known for: moveable type! Xylography is expensive. Although that is just one societal pre-condition for a breakthrough, or in this case: explosion.
For the demand part, cf Lotte Hellinga: "The Gutenberg Revolutions", p207–219, in: Simon Eliot & Jonathan Rose (Eds): "A Companion to
The History of the Book", Blackwell: Malden, Oxford, 2007. And:
The emergence of a literate middle class in the later Middle Ages created a demand for new types of books. These tended to be popular works of a recreational or technical nature, which were often in the vernacular. We know very little about the beginnings of the book trade outside of the monasteries and universities, but certainly there was an independent trade by the late twelfth century. University stationers were free to engage in outside trade and no doubt did so. Books seem most often to have been made to order, but also occasionally for speculation--with no specific buyer in mind. Some orders were large: in 1437 a wholesale bookseller sent an order to a scriptorium in the Low Countries for 200 copies of the Seven Penitential Psalms, 200 copies of Cato's Disticha in Flemish, and 400 copies of a small prayer book.
— Richard W. Clement: "Medieval and Renaissance book production", Utah State University Library Faculty & Staff Publications, Paper 10, 1997. (Online)
Keep in mind that a woodblock in action seems like a great time and cost saver compared to a slow scribe. But that block needs to be perfect. One error and you need a whole other block to start from scratch.
Then there is a slight misconception apparent in how the question is framed. It is correct to describe Gutenberg's innovation as printing (with moveable type). But one of the things he 'stole' to combine it into something fresh was that he took a press from wine making – or even paper production –for his purposes.
This press and its great forces is the biggest cause for wearing out a woodcut so quickly. Using much less force – doing that by hand – is called a rubbing. This technique was apparently used by ancient Egyptians and never ceased to be used on a variety of materials. The techniques of stamping and modeling were also known and used.
That leads to the supply side of the equation. It features the materials needed. Vellum, parchment and papyrus are not really well suited for printing and paper was late to the party in Europe. Earliest book on paper, partially, and imported, made in Europe seems to be the Missal of Silos, dated to 1151.
Second: but they did print with woodblocks before Gutenberg:
A deceased Abbess of Mechelen in Flanders in 1465 had "unum instrumentum ad imprintendum scripturas et ymagines … cum 14 aliis lapideis printis" ("an instrument for printing texts and pictures … with 14 stones for printing") which is probably too early to be a Gutenberg-type printing press in that location.
— WP: Woodblock printing
Or put in another way:
The so-called wood plate printing belongs to the common printing technique of the 14th century. It was popular in Europe until Johannes Gutenberg invented letterpress printing and was used to print any kind of text on paper.
— Ein frühes Druckverfahren – der Holztafeldruck (my translation)
It is just that Gutenberg's inventions (let's call it like that for the purpose of this question) was so disruptive that we now tend to overlook 2 centuries of printing in Europe, because of their small scale.
Exactly dating this block printing of whole books seems contested. But it seems to be at least accepted by some between before 1420 and 1451.
If we had as many early prints as we have Greek pots, we would ont have to guess about practically all the beginnings of printmaking
First dated prints
Few early prints can be dated precisely. A Dutch Madonna has 1418 carved on the block, but the surviving impression is too heavily painted and damaged to reproduce clearly. The next preserved date, 1423, occurs on a south German Saint Christopher, whose drapery flows in the wind as it then did in paintings. (After about 1460 northern drapery straightens out and
and breaks at angles.) The earliest datable Italian print was the subject of a miracle in 1428, when it was tacked to a schoolroom wall in Forli. There it certainly would have yellowed until it was thrown away had not the school caught fire one february day. The crowd that gathered outside saw this paper shoot up out of the flames, hover over the hot updraft, and flutter down into their hands. With cries of "Miracle!" (and who could resist?) the print was carried into the cathedral.
— Alpheus Hyatt Mayor: "Prints & People: A Social History of Printed Pictures", Metropolitan Museum of Art: New York, 1971. (On gBooks)
This is one of the earliest known dated European woodblock prints or woodcuts. It is a portrait of St. Christopher dated 1423 preserved in the John Rylands Library, Manchester, England:
This is of course not the only acceptable view on that matter:
Printing onto cloth had spread much earlier, and was common in Europe by 1300. "In the 13th century the Chinese technique of blockprinting was transmitted to Europe," soon after paper became available in Europe. The print in woodcut, later joined by engraving, quickly became an important cultural tradition for popular religious works, as well as playing cards and other uses.
— Prologue To Graphic Design – The Asian Contribution Woodblock printing>
That means once the demand was recognised different techniques were experimented on to improve on the works of scribes at roughly the same time. Woodblock printing was used quite early on but was quickly diagnosed with all its problems and people realised that for quick, cheap and large volumes, woodcut just wasn't cutting it.
One interesting side note for "why not for [insert your deity]":
The printing trade in Europe was one of the first business sectors governed by capitalist principles, whereas the invention and subsequent development of book printing in East Asia were motivated less by financial interests than religious considerations or initiatives of monarchs devoted to the dissemination of certain texts. Thus it comes as no surprise that the earliest printed products were texts of a religious nature. The oldest woodblock print, a Buddhist Dharani-Sutra scroll probably produced in China between 704 and 751 and measuring only eight centimeters in height but six meters in length, was discovered in southeastern Korea in 1966.
While Arabian woodblock prints are known to have existed since the tenth century, religiously motivated respect for the written word handed down in manuscript form prevented the establishment of printing firms in Islamic countries until well into the eighteenth century.
Book printing was not always welcomed with open arms in Europe. The Church feared that heresies would be disseminated, primarily in vernacular publications. […] It is astonishing to realize that a number of universities initially had reservations about book printing. Although every university city in Germany, with the exception of Greifswald, had a printing shop in the fifteenth century, publications by faculty members were quite rare.
— Urs Leu: "Not everyone was pleased about the invention of book printing. The church and some universities were skeptical." Swiss Nationalmuseum
Addresssing the update to the question:
While woodblock printing is arguably more suitable for the Chinese writing system than the scripts used in Europe, this shows that it is clearly possible to use woodblocks for book printing on a large scale. Saying "woodblock printing is impractical" is not a very good explanation.
Cynthia Brokaw: "Publishing and Popular Literature in Imperial China"
By the 18th century, the population increase and growing demand for texts—and the spread of woodblock printing to the interior and hinterland—ensured the dissemination of a common core of universally popular fictional works throughout China Proper. It was not, however, until the early 20th century and the widespread adoption of mechanized printing, that a true mass readership developed.
Describes precisely why "woodblock printing is impractical". It is expensive and slow and the matrices wear out too quickly under stress. Cost can be driven down with metal while volume increases.
Continuing a trend that began in the late Ming, the cost of cutting woodblocks—by far the most expensive part of the production process—declined, so that investing in commercial publishing became a reasonably attractive enterprise.
For cutting a woodblock you need an expert. For typesetting in metal you arguably need also an expert, but one on a much lower level of expertise. While Gutenberg himself and many other prints after the 42-line bible displayed remarkable craftsmenship, after the incunable era the overall decline in typographic quality for example was readily apparent and only repeated in modern times with the introduction of word processors and DTP.
Comparing the differences between China and Europe, one is also easily mislead by the just quoted article:
Although scribal manuscripts still dominated and continued to exist in China until at least the nineteenth century, China has the longest history of printing in the world.[…]
The nature of Chinese written language limited the practicability and potential use of movable-type printing in China. Since the language consists of thousands of ideograms, grouping and assembling the types is labour intensive and required considerable linguistic knowledge from the workers. Movable-type printing is economically suitable for large quantities for a single run, whereas block printing is more desirable for small quantities produced over long periods of time. The latter was ‘precisely the pattern of book demand and supply in traditional Chinese society’, therefore movable-type did not replace block printing.[…]
Through the examinations of printing technologies, literacy and urbanisation, books prices and access to and control over knowledge, it concludes that the circulation of books in China was substantially smaller than in Europe. It has also analysed the demand for and supply of books containing potentially useful knowledge. The limited data available shows that there was continuity under Song-Yuan-Ming dynasties and that could be correlated with slow but steady technological advance over these centuries. On a per capita basis, growth of book production and circulation declined significantly from the Qing onwards. The decline can be linked to stronger state control over the dissemination of knowledge which can be dated to the early Ming but intensified under the Qing, even though commercialisation of book production expanded under both dynasties. Hierarchy in the Chinese knowledge system is apparent. Books on useful knowledge constituted but a small share of a larger number of books produced, published and preserved throughout the Chinese history. Priority was accorded to Confucian classics and histories. Books may have played a more limited role than was the case in Europe as conduits linking theoretical to tacit knowledge.
— Ting Xu: "The Production and Circulation of Manuscripts and Printed Books in China Compared to Europe, ca. 581-1840", Working Papers in Technology Governance and Economic Dynamics no. 53, 2013. (PDF)