From what I've read, horseback riding began in Western Asia around 6000 years ago, but didn't reach Egypt until about 3700 years ago. Egyptian and cuneiform records go back about 4500 years, before horseback riding had reached that region. Was the arrival of horseback riding to these regions recorded?
Were there any writings where people mentioned that riding horses had suddenly become "a thing," or is it just suddenly mentioned that Ramses had an army of chariots.
I'm not sure about written records, but carvings of horses, most often pulling chariots, appear abruptly in the Near East around about 2,500–2,000 BCE. It used to be thought that this recorded the domestication of horses, but as you say we now know they were domesticated significantly earlier in Central Asia. It appears that the Near Eastern civilisations, who already had donkeys and oxen as draught animals, weren't particularly interested in them to begin with. But then chariots were invented (again in Central Asia), and the horse-and-chariot combination was very rapidly adopted in the Near East. Its depiction in reliefs shows that it became an important part of warfare (and remained so for the next two millennia or so) and a symbol of prestige.