I installed it on another PC and it worked fine.

I dug through the game's files in its installation directory and I managed to solve the issue. I've never experienced anything like this, but here's the solution that worked.

Inside the game's "Content" directory, there's a folder called "ConfigOptions". I think these were used during testing -- they don't seem to be needed at all. However, my computer's name is MSI and one of the configuration files in that directory is called MSI.xml. I checked the file and one of the options it has is "<DirectLoad>Campaign</DirectLoad>".

This is what prevented me from seeing the main menu. It also messed up the saves -- the game *does* save checkpoints, just it never loaded automatically, reverting instead to another part of the game, because of this setting: "<SkipIntroMatch>true</SkipIntroMatch>".

Deleting or renaming the "ConfigOptions" folder fixed my problem. The game now loads to the main menu and I can pick a profile and play the game, access Versus Mode, and so on.

So if anyone runs into a similar problem, this is the fix.