I believe the pacing of the game is well thought out, and I certainly felt the panic in the final minutes. I think it would be possible to adjust some of the timing and shorten the cooldown, but it isn't really necessary. I'm just used to jam games where you usually try to get the player's attention fast and then keep them interested all the way through, which works well for a lot of arcade style high-score games that people make. I do agree with your last point, that maybe it is a little harsh if the player loses towards the end of the 15 minute cooldown, but I think the longer, slowly increasing tension style of pacing works extremely well for your game.
Overall, I think it may be beneficial to shorten the length of the cooldown and give the player slightly less time between critical tasks, but not so much that it's hectic from the start. Even if it only cuts out a minute or two of the game, it could make the game more forgiving.
Those are my thoughts
Hope you find them helpful :)