It was about myth. Anglo-Saxon myth was horse oriented. Their legendary leaders during the Saxon invasion of Britain were Hengist and Horsa, twins whose names are both words for stallion. Tolkien designed the Rohirrim based on an idealized version of these myths and the culture of ancient England. Anglo-saxons were sea people (interestingly there's always been a connection in Indo-European myth between the sea god and horses, compare greek Poseidon and the strong connection between rivers and "river horses".) Tolkien simply used the idea of Saxons from a land locked area as inspiration for the Rohirrim. There are clearly many aspects of Rohirric society that are derived from Anglo Saxons (burial practices, nobility, combat) but it's impossible to say how much, as Tolkien himself pointed out that the Rohirrim were NOT anglo-saxons in all ways. Two things we know about Tolkien: he was the chair of Anglo-Saxon studies, and he wanted to create an entirely English mythology based on real myths and legends. No surprise this material turns up in his novels.
Anglo Saxons had cavalry. The Norman French had THE cavalry. The last king of the house of Wessex had died childless and he was succeeded by Harold Godwinsson amid much controversy. Everyone wanted Britain, and because of the Heathen invasions of the Viking age, there were legitimate claims to the throne from Scandinavia and Norman France. So Harald Hardrada of Norway invaded and was defeated by a hastily assembled English army. Unfortunately for Harold, William the Conqueror invaded immediately afterward with a well planned and organized campaign and defeated Harold's army at the battle of Hastings in 1066.
The English had successfully and repeatedly repulsed almost 300 years of Norse invasion. They did this using strategically placed fortresses called "burhs". Since most of the Norse raids were for plunder and slaves, the English would enclose entire cities worth of people and stuff in these burhs. The Norse, too far from home for reinforcements, were unwilling to lay long expensive sieges. Harold attacked William to try and contain him in the beachhead. He meant to raise armies and bring them south and probably would have used the same burh based strategy while trying to cut off the supply and reinforcement lines to the Norman army. Defending an island is a huge advantage. The invading army must supply itself from the surrounding land and either transport siege machines or build them on a foreign shore.
So it might have been a popular romantic thought (and romantic though was varied and bizarre at the time) that cavalry could have changed the outcome, but it seems unlikely from what we know. Harold lost because he'd just fought a huge pitched battle and never recovered.
While William was a descendant of Rollo, the first Duke of Normandy, a viking who was given extensive lands in exchange for basically not destroying Paris, the invading were French. They spoke French and were integrated into French (i.e. Frankish) society. Harald Hardrada was a viking from Norway. Harold Godwinsson was Anglo-Saxon.
The Uffington horse is modern. Tolkien did say that Theoden's banner was inspired by the White Horse, but that's really the only connection there. It's no stretch to think that such a culture would use horses on their banners.