That could also work with savegames, in that you can have saves, but make the default on startup be to restore where one was in the last game. Many games provide a “continue” option at the top of the main menu, I think reflecting the fact that that’s what a player wants to do 99% of the time.
Two caveats:
If it’s an action game and there’s loading involved, it’d be nice to know when the load is done, since you may immediately have to be reacting to something in-game. I’d rather have it attempt to load the game and then go into a “pause” mode, maybe with some overlay or something indicating the current game state (like to remind you what level or wherever you are).
It’s possible – because we live in an imperfect world with imperfect software – for save games to get into a broken state, and if so, you don’t want to make it impossible to reach the main menu if trying to load the last save game is crashing the thing. Maybe make the game detect that the last load failed, akin to web browsers, and then head to a menu in that case.