You can run without a thermostat, though it would take allot longer to get up to temp + cause it to run cold more so on free way driving. Taking it out is a quick / easy check to do.
Bleed the system, but you'd have a bouncing idle as soon as any form of temp gets into the car.
Is the radiator cap common to each radiator?
If that doesn't work, get your radiator pressure tested + that check they do to detect oil in the coolant.