Something is going on with that cooling system. Unusual overheating is very bad, that's the big worry ultimately. But the constant air in the system is a symptom of something else, and that is probably causing the overheating.
Either there is a leak that you haven't found yet, causing the excess air, or something simple like the cap is letting air get in regularly. You better hope it's the cap or a gasket like the intake or thermostat housing, because the other possibilities are bad, such as the head gaskets. With my Explorers, when cold I can always open the radiator cap, and the coolant is right up to and inside the cap.
I'd keep watching everything for anything new or different, while still working to get air out of the radiator. The method of "burping" it has nothing to do with the excess air, or the overheating. Under normal operating conditions, there's nothing special to do in filling the radiator fully. These are not weird like a Subaru with various coolant hoses higher than the cap or fill point.