Suddenly the music stopped playing and the Narrator stopped narrating (I have the subtitles option on and they were stuck). Just after, during a boss fight the game slowed down, almost freezing. Still with a super-lagged game, I managed to get back to the Bastion and exit the game. Then I quit the game, but the screen went black for ages, so I eventually turned off the PC manually.

Next time I start the game, there is no Continue option!! My saves seems to be intact in the Saves folder. I tried to repair and uninstall/install. No luck.

What can I do?

Maybe you can upload the save files and let someone test it? So we can figure out if the save files are damaged or not. If the save files are ok then it can be the game being buggy.

somehow, at like 80% completed, my savegame got corrupted

it's a game I'll never finish because of that, and leaves a shit taste in my mouth because if I'm going to be forced to not be allowed to save, this is something that should NEVER happen

meh it was tiresome and repetitive anyway