My observations so far agree with OP.
Shield wall works well, it's a defensive formation that's good for pushing up to the enemy. In Captains Mode it's pretty much a must when facing against enemy archers and will let you push up into the using either Hold Position or Advance (as Charge will remove the formation they were in) but as it's been mentioned it really only works against cavalry if there are atleast three ranks of men in it.
I dislike the Square and Circle formations, they feel lackluster. Both seem to operate similar to a shield wall but I would've assumed the square would be atleast a little bit more aggressive when dealing with cav, unfortunately I've noticed that the AI is slow, they will often have a staggeringly long delay between a horseman riding into them and actually attacking. Sure, they will take out a lone soldier in the middle of the ranks but that's because they've become a speed bump and actually slowed them down to a crawl/stop to actually hit them.
Circle not only has this problem, but I'd expect it as I've always assumed the circle is a defensive formation, but their problem is that the troops don't raise their shields high enough, they all keep them chest height which means they'll constantly be shot in the back by horse archers that have them pinned, they don't cover one another like they should and so I've found having a dense shield wall is a far better option to do while moving your units back away from the HA as best you can.
Personally I'd like to see Taleworlds have a look to see if there is something wrong with the AI at the soldier level and possibly see if they can change the AI blocking with shields to actually be aware to raise their shields higher.
On the topic of bracing spears, it's really simple, I don't think we should see lawnmower phalanxes running around as it'd probably be game breaking but stationary units should be able to brace which would be a sort of reverse of the couch lance that really takes into account how fast the enemy is approaching to cause damage. The faster the enemy moves the more damage they take, obviously. A horse running at full speed into them will get demolished but infantry or horses moving at a snails pace will feel a prick at most, meaning that a smart player will know when to brace and went to not. If it's possible, I would lock players turning ability while in brace to like 45 degrees at most and put in a setup/cooldown period so that a player can't just do a 360 spin and kill every single horse running at them, the setting up and cooldown period meaning they can't kill, do a 180 and brace against for the second horseman riding at them in a few milliseconds.
Edit: obviously once a braced spear has hit something, it should come out of brace as to avoid the foreseeable issue of players/soldiers getting stuck on the spear and just repeatedly bumping onto it until they die. The way I see this ability is that it's a primarily anti-charge defense that is exposed on the rear/flanks but also lowers the ability to block arrows so players that just sit in a braced position on the point will find themselves getting done in by archers or infantry pushing into them with their shields up.