Josh, these guys here have mostly never heard of burping a cooling system. They work on older iron when the guys who were designing it didn't know of a "better" system like the newer cars.
I have had to help burp some of the new stuff by parking at an angle, loosening certain hoses, that sort of thing.
With these, you just fill it, run it, and refill if needed.
I have had two water pump failures that caused symptoms like you described. That is in 40 years of normal failures, so it ain't common.
One, was on my old Jeep. It would run hot at speed, cool off at idle. Normal stuff did not help.
Finally, after a couple of weeks, I was sitting at a red light, and my fan started hitting my radiator.
Got out, and the water pump shaft had slid out of the housing far enough for the fan to hit with no water leakage at all.
I was about two blocks from help, so I just slid it back in and drove it back. Turns out the impeller was loose on the shaft.
At low speeds, it would grip enough to pump, but when you tried to force it to pump faster, it started slipping.
The other one was a little different. It was on a Pontiac. The pump started leaking, so the guy replaced it.
After that, it never cooled right. Faster you drove, the worse it was.
Finally he came to me in desperation.
Standard procedure. If it was working before you did something, and not after, check what you did.
No matter how much you revved the engine with the cap off, I never saw much flow.
I remembered the loose impeller, so I pulled the new water pump off.
I could not turn the impeller on the shaft, but it had a warranty, so we took it back to the store. He agreed to replace it, and asked what it was for, since we did not have the box.
He looked it up, brought out a new pump. We took it out of the box intending to use that box to put the old one in, and it was real obvious that they were different.
Housing was the same, fan mount was the same, but the impeller was totally different.
As soon as I saw it, I realized that the other one was wrong, It was much smaller than the hole.
He checked a few others in stock and they all had the same impeller.
Only the one he got first had the small one. It seems that the rebuilder had pressed on the wrong impeller, or it was for a different model car.
Replaced it, and all was well.