I had started to write something about this earlier, and then forgot I didn't post it, so it does look like I pulled that number out of nowhere.
In the turn 1 kingdom, I'm pretty sure the first player can always win on turn 2 if neither player wins on turn 1; the setup of the Mission turn is simply a topdecked Fortress, which can be done for $2 with Travelling Fair + Alms (so, always). The one potential hangup I see is that the second player gain multiple Villas somehow and keep you from having enough Buys, but on the other hand turn 2 has the advantage of being able to buy cards normally, so you can likely get around that somehow.
So the probability of winning is 5/12 + 7/12*7/12 (win first turn + don't win first turn and neither does P2) = 109/144 > 75%. (And then divide by 11/12, 109/132 is about 82.6%.)
The 83% is still too high though; I didn't account for the 1/12 case of P2 having a 5/2 and splitting the Squires evenly. P2 might have a slight advantage here (except in the case where P1 also got a 5/2), but it's certainly not a lost cause for P1. So, somewhere between 78% (if 5/2 is 50-50) and 83% (if it were always a P2 win), but closer to the lower end of that.