Variant Friendly Timer Rule:
Start the timer when the next to last person has grabbed their turn tile.
If the ships are all equally well-built, I think the first player has quite a large advantage, so my concern would be that, under your rule, all of the remaining players might decide together to take a very long time in order to optimize their ships as well as possible to overcome this advantage.
My impression about the game is that the "1" tile is worth too much, which itself creates the time pressure, regardless of what you do after that. I'm not sure what I would change to compensate for that. Maybe increase (double?) the number of days of delay for the various events?