Yeah my wife and I have played this twice (2P) and lost both times - got very close the last time but my wife got incredibly unlucky with a chain of restrained cards when we just needed to get to the boat.
I definitely see investigator selection as one of the most important parts of the game and would never randomise it completely. We normally spend a good 10 minutes planning a well rounded team - for this scenario I think you definitely need influence and I've found the athlete is really good in being able to move around the map pretty swiftly (also pretty important 2P).
That's definitely a great point. I don't think I'd ever go completely non-random as I love doing so in every Arkham game I play, but in a 3 player game of Mansions I think in future we'll select 2 random characters, and 1 proxy "best" like Rita Young as you and Rick suggested to make sure the game doesn't hit any flow ruining stumbles.