Depends on the game, if its a fighting game(with reaction based stuff, which is getting rarer and rarer now) that requires precise inputs, local is the only way to go, if its something slightly more casual it depends on a case for case basis.
Something I really liked with the DS (3ds also had this feature, but much less utilized) was download play, that way you can get the benefits you are mentioning about screen rendering and stuff like that, and the players can still play it locally for the same cost as 1 game.
I like teambased onlinegames in theory, but in practice it's just hard to set up a team with a similar goal when playing so mostly end up soloing which could be switched for a better experience elsewere.
I do absolutely not like the splatoon approach, where they sell a full priced game, that is basicly only online, and teambased, but it marketed as "competive team game" and still dont have a competent local multiplayer (no, 8 wii u's should not be required for a local multiplayermatch).