Sounds like a proper piss off alright.
You mean radiator cap right??
The shaft seal on water pumps have been known to allow air to be admitted when operating if the seal seat is not installed 100% square to the shaft (causing it to wobble) or the seal is damaged or faulty. A pump in this condition will pass a pressure test for external leaks.
I can't say that I have ever seen a heater tap suck air & not leak, but there is always a first time. It is a high point on the suction side.
I was also wondering what pressure your system holds with the cap on, and what pressure it is when it starts to push coolant to the overflow, as usually once pressure builds high enough then air can't be drawn in (theoretically). Might give us a clue.
The other thing that might be worth a look at is the radiator cap itself. Some cap designs have a float type arrangement for the recovery button (valve) & other have spring type recovery button. The float type gives hit & miss sealing if not completely full of coolant.
So air drawn in when cold (when running via pump seal?) and maybe cap not holding pressure because of said air??. The recovery line from radiator cap should also hold a vacuum. I take it makes air when cold or hot and it does not recover coolant from the bottle by itself as it cools when this happens.
Also good to here the performance at a more desirable level.