It's important to note that concrete information on how shields were used is scant, so a large part of any discussion on this subject is speculation and logic.
That said, kite shields had an obvious advantage in extending protection to the lower half of the body. This was especially relevant to the cavalry, and particularly so in a period when leg armour was not common, since the height of their legs make them natural targets for an enemy. Of course, the downsides of kite shields are equally obvious: they were heavy and awkward to maneuver.
That is, the benefits were less meaningful, and yet the drawbacks more serious, for leg infantry.
Round shields did not actually "fall out of use" until the Renaissance, and they also continued to be held in a central grip. It was the weight of kite shields which necessitated strapping them to the users, rather than inherent potential tactical advantages per se. Later manuals indicate that the smaller, and lighter round bucklers were extended in a fist far out from the body, so as to be ready to block any incoming blows with little movement. Strapping them to the forearm would've considerably limited this.
See this depiction from the late 13th century Boulogne-sur-Mer BM MS.131:
Anyway, returning to kite shields. It's probably a safe bet that it was not a coincidence that kite shields became common around the time cavalry began to dominate in Medieval Europe, and declined once leg armour became widespread. The heater shield that largely replaced them are noticeably shorter. In other words, once the benefits for leg protection diminished, kite shields were replaced by lighter and more maneuverable, shorter shields. Note that round shields remained in use throughout the same period.
For illustration, consider the knights with kite shields from the Bayeux Tapestry:
Compare and contrast with these from the Rochester Bestiary, circa mid-thirteenth century:
In the latter, we can see the legs were protected by chausses. This is conspicuously absent from most of the armoured figures depicted in the Bayeux Tapestry, most of whom only wore knee-length hauberks.