While the shortening of the work week from the late 1840s onwards was certainly a factor in the rise of modern football (or soccer), it was only one of several changing circumstances which allowed the game to flourish. Further, football was but one of many leisure activities which benefited from a shorter working week.
Also important were factors such as increasing urbanization, the expanding railway network (making it easier for teams and supporters to travel), professionalism (making the game a possible source of steady income for those who could not afford to play as unpaid 'gentlemen'), education and the creation of a sporting press, and the national standardization of the rules of the game. The latter, in turn, allowed the emergence of national competitions such as the FA Cup (1871) and the English Football League (1888), though the former in particular took a while to gain popularity with clubs.
A number of sources have noted "the connection between the rise of modern football in the UK and the shortening of the work week." From 1850, a series of labour laws led to an increasing number of workers having Saturday afternoons off, creating circumstances which facilitated a rapid increase in the number of football clubs (among other sources of entertainment).
Until the last quarter of the nineteenth century…industrial laborers
had neither time nor energy to pursue recreational activities. Not
until several mid-century Factory Acts limited the hours of women and
children in textile factories did the traditional six-day work week
for adult males begin to be shortened. By 1870 a half-day Saturday was
realized in most factories, mines and workshops, providing leisure
time for the bulk of the laboring population.
Source: William J. Baker, ‘The Making of a Working Class Football Culture in Victorian England’
As an example,
It is no coincidence that the Factory Act 1874, which guaranteed a
Saturday half-holiday for textile workers, was contemporaneous with
the sudden growth of football in the Lancashire mill-towns. There was
now a convenient, shared time for this popular recreation, whether as
player or spectator.
Source: Lee Jackson, ‘Palaces of Pleasure: From Music Halls to the Seaside to Football, How the Victorians Invented Mass Entertainment' (2019)
This is not to say that there weren't football clubs before the 1850s. Rochdale, for example, had at least four clubs in the 1840s. Clubs had many different origins; some were founded by businesses/workers (Arsenal, Man Utd.), some by pub landlords (Chelsea), some by cricket clubs (Sheff. Wed., Preston NE), some by churches even (Bolton, Aston Villa). This growth was uneven, though. Note, for example,
the experience of Liverpool and east London where dockworkers in
particular and workers in general were late in gaining their Saturday
holiday; local football leagues were slower to develop in both areas
by comparison to central Lancashire or south Yorkshire where the
half-day holiday was instituted a decade or so earlier.
Source: David Goldblatt, ‘The Ball is Round: A Global History of Football’ (2008)
But it wasn’t just labour laws and more free time which led to the rise in the popularity of football:
In the same year  Parliament passed an Education Act, ordaining
that a school should be placed within the reach of every English
child. As English schoolboys learned their three R 's, they also
learned to play organized games. In providing reading and writing
skills for the children of industrial laborers, the Education Act of
1870 created a mass reading audience for the sports press which
burgeoned in the last two decades of the century and inadvertently
placed a football in the hands or at the feet of every English
The expansion of the railway network also contributed, though initially more for teams than for their supporters:
The industrialization of transport technologies and infrastructure
underwrote the increasing size of crowds and the enlarged geographical
scope of leagues and cup competitions. Between towns and cities, the
rail network was now substantially complete and reasonably priced,
although the Football League could still exclude Sunderland in 1890 on
the grounds that the cost of travel for other clubs to the far
north-east was prohibitive. Trains provided the means for the bigger
teams to conduct national Christmas and Easter tours to top up their
coffers and for international teams to meet in the annual Home
Countries championship, but they were not as yet being used by
spectators. Aside from very local derbies away fans were almost absent
during this whole period (1880-1914).
However, it wasn’t just among labourers that the popularity of football increased:
…during the mid-century, it would increasingly find favour in the
public schools, prized as a character- building sport. Amateur clubs
then began to emerge, principally composed of middle- class men who
had enjoyed playing the game in their school days.
There remained, though, a major impediment to the long-term growth of the sport into the modern game of today:
Football matches were generally played according to local custom and
practice…. Pupils attending other schools, therefore, played by their
own local rules. Rugby pupils drew up a written code in 1845 for their
own benefit; rival educational establishments continued with their
separate traditions. Eton’s ‘field game’, in particular, did not
permit handling of the ball (although it did include ‘bullies’ and
‘rouges’, similar to scrums and tries in the Rugby game).
Even fundamental rules, such as the size of the pitch and the number of players per team could vary enormously from district to district. Thus, if two teams from different areas wanted to play each other, the rules had to be negotiated first. An attempt at some bringing some uniformity to the rules was made in many local areas in the 1840s (for example, in Rochdale and among some parishes in Warwickshire), but this was still far from being national or even county-wide.
Also, at Cambridge in 1848, several schools got together and came up with the Cambridge Rules, but these conflicted with those of Rugby school (and others) and, ultimately, this led to the splitting of football into association football (or soccer) and rugby football (which subsequently split into Union and League). Ultimately, it was the Cambridge Rules, together with the Sheffield Rules developed from 1858, which became the basis for the laws of the modern game as administered by the Football Association (FA), founded in 1863.
It took a while, though, for these rules to become national (and for truly national competitions to emerge), not least because the FA leadership was dominated by ‘gentlemen of leisure’ with, as yet, little appreciation for the vibrancy of the game among industrial workers. Further, the issue of amateur vs. professional players caused divisions in these early years (but did not end up with association football splitting in the way that Rugby football did) until the Football Association caved in on the issue in 1885, thereby opening up the game as a paying profession for thousands of young men.
John Goulstone, 'The working‐class origins of modern football' (2007)