Yes, I struggled to find a big role for cavalry even as Takeda. The problem is the mass of the enemy, at least early on, are cheap ashigaru which are lethal to cavalry close up with yaris or at range with bows.

If you have time to micro it, the "Benny Hill" tactic of luring away spears to chase your cavalry could work - shooting them down as you do so. However, it is a bit of an exploit and rather hard to pull off in a proper battle when you are busy, given the very fast movement rates in the game. Bow cav may be best for that, given they can skirmish. A variation on this might is to lure out enemy cavalry into a premature charge into your spears - however again it takes quite some micro. Trying both things early in the Takeda campaign, I would often mess up and lose my lure due to inattention.

One thing I've started doing more while playing EB and the Third Age (where units have quite high morale) is double teaming the morale bomb charges - have cavalry stacked one behind another. IIRC the usual hammer and anvil tactic of earlier TW games is not quite as effective in S2TW as in earlier vanilla TW games, so a double tap may help. Waiting until the committed enemy infantry is no longer eager or steady may also be wise. Hiding the cavalry prior to charging should also help the morale bomb.

I have not done the maths or tested it, but I suspect Katana cavalry are superior to to yari for countering enemy cav - especially if you can't be relied upon to get a proper charge in.