Cavalry has 1 massive advantage over all other units - mobility. A unit of No-Dachis, apart from the whole getting-shot-to-pieces-before-they-even-connect-with-an-enemy, simply cannot threaten as many points in the enemy formation as a unit of cavalry can as they're much slower and thus much easier to zone. A group of Yari Cavalry can threaten the entire enemy rear from practically any position behind his line. This also means that they naturally get weaker as they get more numerous, since you don't need 4 squads to hit 1 weak point - 1 will easily suffice with perhaps another to threaten a counterattack at a new point if the enemy tries to rearrange his spears in accordance to your first charge. This is also why I always try to rush a stables for Light Cavalry as there is simply nothing in the game that can fill the role that Light/Yari Cav can (Katanas are way different as they're slower but can dominate a tough melee against non-spears). Its actually pretty brilliantly balanced and Cavalry feels very much in place in Shogun 2 compared to something like Rome, where Cataphracts/Praetorian Cavalry could counter absolutely everything in the whole game (except archer chariots but thats a whole other discussion) due to being practically free and virtually uncounterable for non-Egypt/Brittania on top of being lightningly fast for a muscle unit.

I played an all-cavalry army as Takeda on Legendary, relying on primarily Katana and Bow Cavalry with a few Yaris and Fires to rout weakened units and hit enemy generals, but I predict no future for it as its both incredibly tedious to micromanage and simply ineffective if the enemy has more than 1/3 of his melee made up of spears or Naginata (fewer than that and your Bows will pick them apart). More often than not my Katanas ended up jumping off their horses and charging in, so I ditched half of them and replaced them with Katana Samurai and it made things ridiculously easy compared to before. The primary advantage of an all-cavalry army is on the campaign map, as a 40%+ movement speed-general + nothing but cavalry can cover more than twice the ground that an infantry-army can. This is actually pretty huge if your stack can fight a defensive battle against the enemy as you can easily swing into favourable positions or pick off reinforcing stacks completely safely. I highly recommend it (maybe on hard if you're not used to cavalry) just for the experience it gets you in terms of campaign map positioning and stack manipulation.

There is no faction that I'd ever completely ditch Cavalry with. There is nothing that can fill their role. Any morale-reducing action will triple in effectiveness if followed up by a cavalry charge and the threat of this alone is worth at least 1 unit slot.