For anyone who made the same mistake I did: You can use a key on any lock on a door, at any point. I found the game pretty much impossible when i thought you had to wait for timelocks to disappear before you could open a regular lock on that door. (Just me? Oh well.)
Once I got past that, I won the game on the first try, but it still felt like an accomplishment and not something trivial. It's quite good.