I'm glad they didn't (and am fairly certain that they'll drop the party mechanic when they remake V if not earlier assuming the next game is something else) but let's see...
Ys I really should stay a solo adventure if only because of the ending but your idea works. Have them together to explore the tower and they could insert a few simple 'put this character in the lead to proceeed' puzzles.
Ys II: Umm... dunno really. I suppose Lilia could tag along and get abducted, sick or petrified every ten minutes. Wait a minute...
Ys III: We'll pass on this and just skip to Oath. Dogi tags along and replaces Earth magic with his mighty fists. As a postgame bonus, Elena lives up to that official art of her, takes up the Brave Sword and becomes your third party member.
Ys V: Terra follows you around more than she does already and her AI is programmed to try to steal all the jewels the monsters drop. To make her worthwhile, those gems you keep can be sold for a higher price than normal. Probably throw in a few areas where she can pick locks to open doors/chests too.
Ys V: See Ys V (and substitute Emel), then add Olha for certain segments and give her mechanics similar to how Aisha played in Seven. Geis could similarly be an endgame character.
Origin: Massive party mechanics allowing all those cool characters Falcom created to actually do something. As a bonus, by beating each scenario you unlock a set of characters who can be freely assigned to your party in the Arena (Knights, Mages/Priests, Clan of Darkness).
Origin wouldn't have worked on the party system. Conceivably, you could play as Yunica and climb the tower with Gemma, Cadena, and Miuscha after while, but she climbed the tower originally out of impulse. Alone. Hugo and the Third Character...wouldn't work.
Napishtim actually has quite a few named characters who could fill the role. Ord's kid with the funny name...Ur or something like that? He could travel with you. And there's the other Rehdan who lives in Rimorge that crafts the weapons...Leav or something? I don't remember, but I think she was a warrior. That could work, and she could also temper your weapons anywhere to save you backtracking to Rimorge. And then there's pirate girl...I'm not opposed to Geis, either, but the point here is there are plentiful options. I completely see this game working on the party system at face value, anywyas.
Go to Gamestop, buy a Steam card, and use it to purchase Ys Origin. You won't be disappointed. For more info: http://www.worldofys.com/yso/
They confirmed that a boss rush mode is in the game and will be unlocked after beating it once. How exactly they're going to make it work considering all the variables the party system introduces I have no idea but it's going to be there. We'll know in a day or two.