You're worrying too much with this. Just drive it, park it, let it cool, fill. Few times and you'll get all the air out. Never really had any success with 'burping' it with the cap off and the while thing running. All I do is end up spilling more coolant...
T-stats are cheap. They're also crap. So like 'moonshinefuel' stated, replace it. Easier now than any other time (before it's 10 degrees out!). You should be at 190 anyway. These things don't last forever.
The radiator cap is already at the system's highest point, so air will go there once the t-stat opens while running.