They put up a great fight and the initial Norman attempts were unsuccessful, but at the end, the cavalry prevailed.
I’m also not sure there were many horses involved, though tbh it’s been a couple of decades since I cared about this battle and my memory may be poor.
Harold Godwinson was an extraordinary commander. To permanently throw the Vikings out of Britain and then march and almost repel the Normans (though basically also Vikings) shows terrific determination. And he chose his territory well: he had the upland position boxed in at the sides and with the Normans having only the sea at their back — he should have won. His troops were incredibly tough guys.
Had he carried the day I think he would have been a good leader. His case was stronger than subsequent propaganda has portrayed.
IIRC the Norman cavalry played a decisive role in routing the fyrd at least, but it has been some years since I looked into that as well, so my memory may be faulty.
Harold might or might not have caught an arrow in his eye. There is a great deal of symbolism in that narrative, because an arrow in an eye would be a divine punishment for oathbreakers, and William insisted that Harold previously took an oath on holy relics to be his liege.
So it is not impossible that the detail about the arrow in eye is a latter embellishment to make the story more righteous from the Norman side.
He immediately said “arrow to the eye”! So for his sake at least I hope the story is true.
Edit: Wikipedia has a good overview if you Google the term. My guess is that proper horse mounted combat requires far more intensive and consistent training that could typically only be undertaken by a noble or knightly class with no other real responsibilities. Whereas anyone with access to a horse could fulfill their (rarely exercised) martial duties as infantry.