I have taken a closer look into the matter of the kite shields. It is tricky. You are right saying the round shield was dominant in the 10th century, while the 11th century saw the kite shield. The problem is: there is nothing in between.
Almost out of nowhere, kite shields pop up, fully developed, and are universally used in the 11th century in western Europe, be it Spain, France, England, Germany or Italy. The Eastern Romans used it, and even the Islamic forces to a small extend. Yet there seems little to no transitional shield between the round and the kite shields, unless you count the sparse evidence for oval shields, including 10th century Germany. You see our Franci equipped with them.
The very first depiction (there are no surviving examples) is in the so-called Farfa-Bible (Cod. Vat. lat. 5729), which probably dates around 1000/1001 and cannot be younger than 1047; its folio 145 is the best image but I cannot find this online. By the mid-11th century, there were only few if any round shields depicted; in the German Codex Epternacensis (ca. 1030-1050) there is not even a single round shield!
It is subject to debate from where these new shields originated. Considering the Franco-Spanish evidence being the earliest, and Italy being late to the show with the first kite shields appearing only in 1070, but Byzantine art depicting them as early as second quarter of the 11th century, neither a west-to-east nor an east-to-west line can be drawn.
The shield itself, as many have noted, is perfectly suited for cavalry combat, shielding the left completely when necessary, especially during the charge. This was the ultimate shield for charges with the lance couched. Still it allowed for flexible use in hand to hand combat.
See attachment for nice picture of the advantage offered by the kite shield: