Beacause Japanese had really small, poor horses and themselves lacked many horsefucker innovations to make horse warfare work.>pic related
Here's your non-horsefucker horse of the place. The folks in the photo are 1.5m or so.
In turn, Japanese horses, when armored up, couldn't really do much running at all. They also lacked horseshoes (Japanese used rice straw shoes in tough terrain, you'd need to carry several for the horse), their saddles were made completely of lacquered wood (heavy) and they did not breed (to get them bigger), nor gild (to make them actually follow orders against their wishes) their horses.
In turn the horses walked side by side with the infantry, effectively being more of a heavy infantry for plugging holes (when they got there, most people would actually even dismount) as they develop since they could do a very short gallop to safety if things turned to shit. The other use was running down survivors during a route, though obviously, that's not quite as easy in the Japanese terrain as it is in Europe.
See this Koreashit movie for some context of how such a chase in a Japan-like terrain actually looks like (minus the obvious uberarchery bullshit).https://www.youtube.com/watch?v=DlETvAfOMbk
This wasn't completely universal, though. Touhouku (Date, in Shogun 2) had some basic breeding going and a decent reputation for actually being better with their horses. The folks actually have a recorded charge in a few instances. Though you need to keep in mind the previous bit about just how utterly sad the horse situation in Japan was, so don't expect these folks to suddenly have throughbreds in full armor, triangle formation doing a LotR style charge which decides the battle. They would actually only get some 5-12 horsemen and run it into a small group of soldiers to quickly overwhelm them.