Fastest Escape *does* work! If you can't get it it's because you're not fast enough. Yes even if you got through in a few seconds, that's still probably not fast enough.
For reference, your score decreases as time goes by, so it depends on having a high enough score. It unlocked for me with a score of 9775. I believe when I had a 9600-ish score it did not unlock (because that wasn't fast enough).
Also, yeah, you gotta already know a certain thing (which you find out by playing the game normally) if you want any shot at the Fastest Escape medal.