Timed door closes too fast even in training mode, yes even when holding Shift or even spamming the Shift key for those little boosts.
Should have tested better before release.
As-is this game is probably not beatable, especially if you require those timed runs in some levels which I'm guessing you do otherwise why would it be in the tutorial stage?
I believe you but can't reproduce the bug on my end, it's probably an edge case that affects a small amount of people and, being an old game now, not worth the time to go back and fix.
Good news is the rest of the game does not rely on this timed door mechanic. It was just to teach you how to dash, which is useful for dodging bullets in game. You can play the levels without beating the tutorial.