It could be that the primer bulb is cracked rubber and not doing its job. It could also be that the bowl nut (jet) has varnish in it, which is more likely.

From time to time the question is asked about why an engine will not run long, or smoothly, or will start with starting fluid and not continue to run. The problem is usually because of a blocked or constricted jet. The small ones are easy to miss.
Shown are 3 of the typical jets and the small holes. All 3 are Tecumseh jets. Briggs jets are similar with the center-bottom hole and one on the side. All these holes need to be fully clean and open for the engine to run correctly.
Some jets only have the center-bottom hole.
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The jet on my mower looks exactly like the one in the first picture. I did thoroughly clean it and I thought that would solve the problem, but it didn't work. The primer bulb appears to be just fine. I also checked the float. Maybe I just need a new carb.

The bulb can look good but still be cracked in the back where it seats against the carb. Also, on your model, the bulb has to be fully pushed in, unlike the Briggs, which I think are better.

Look in the end of the carb and see if you see gas being pushed up through the jet.

Even if the bulb doesn't work correctly (and it probably doesn't) it wouldn't explain the necessity of priming a hot engine. That is what puzzles me the most. Would a new carburetor fix this problem?

Naww, it happens to a lot of them. I've found it is just how they work. Push away, get it primed, and it should start. Those carbs do not have a fuel pump, is why they have the primer bulb.
To remove the primer bulb, where is a thin, tin snap ring down around the bulb, with little feet that grip the side of the hole. Using a small screw-driver, carefully pry this ring out, straighten it out for re-use. Remove the bulb. Carefully clesn out any dirt under the bulb. Place the new bulb down in the hole, then with a larger flat screwdriver push the ring down until it is back down in place!
That little bolt that holds the bowl onto the carb. You should send a puff of air thru it after sticking it with the little wire. Just a puff by mouth! Gets rid of the stuff you poked out of it. Oh, go on, it won't kill ya--I've done many that way!

Do you know the compression pounds in the cylinder? Also, the flywheel key could be sheared just a tiny bit.