I tried doing a run today of Any% No Level Skips but I couldn't start a new story mode, how do you do it?

Have you tried opening the \Steam\userdata<steamid>\238460\remote\BattleBlockTheaterSave file and write something in to corrupt it? Then launch BBT and it should tell you the file is corrupted and offers to create a new one.

I just realized that other runs on the site dont start with a reseted profile, so I guess I don't have to. But I'll try it out!