If you are going to use it off road at all, I would 100% recommend a cooler.
Yes, it would eventually work all the air out of the system, the more you use it. If the pump didn't squeal when you first started the engine, then it must have been primed good enough. I would still check the reservoir fluid level everytime after you drive it for the first few times, just to make sure you didn't have any air pockets in the system.