I've always understood the point in Horsearchers in Total war and many other games, that they can defeat nearly every other unit when microed correctly, but can be defeated by most units if not.

What I find rather funny to do with cavalry is to take undefended settlements^^, a squadron of 6-8 cavalry(2-2-2 + general + Guard^^) units can defeat an AI driven Garrison with few problems. Not something I'd advise you to do when things are getting tight, more like an alternative to marching your Uber army all across Japan.

Generally I like to have the aforementioned 2 units, however I prefer to have them in addition to my general, as a very bad experience lately made me more cautious when using my general :/ Normally I use them to crush the enemy cavalry first and then charge the enemy from behind or Distort enemy archers. Dunno if it works on legendary but I never had problems with this agenda on normal(also in ROTS).