Deadly, you're mostly right, but I'd say it in a very different way:
If you don't mind player 1 accidentally/intentionally able to mess with player 2's controls during P2's play, and vice versa, you're fine hooking both to the same inputs for alternating play games. Plus, most games are set up default so the players have to swap, or have it wired like you want.
However, many people don't like having one player able to mess up the other's game, so they wire the players separately. The problem with this is that unless the game has a seperate player control mode, separating the controls might not work. And many games that have a separate controls mode, that mode is cocktail, so unless player two can play upside down, you need a special mame that doesn't flip player 2 in cocktail mode.
So the easy answer is, for the OP's situation, wiring them together like the OP said is the easiest way out.