On that note, a diagnostic question/thought:
Why was it hard to start all the times after it had been running fine but sat for three minutes, hot?
Now that does sound a little heat-soaky to me. Water stops going through the jackets, stuff on the outside of the motor gets hotter.
BUT: it's also true that those older strokers are all a bit individualistic as far as starting procedure. The motor on the back of the boat I took out after abandoning you to the mercy of Neptune there can be started in a maximum of two pulls at any temperature...IF you have the throttle and choke positioned correctly for that temperature.
If you choke it too much when hot, it fouls and will fight you tooth and nail; you may not be able to start it without removing the plugs. It took me probably a hundred repetitions of starting it (which probably included at least a thousand pulls, likely more, beause the learning phase involved a lot of failure) at various temperatures before I got to the point that I felt I could always start it. But now I never think twice about it. If it's been sitting for a few days, two pulls. If not, one and done. This same motor gave me FITS before I dialed it in.
The previous engine, which John has seen photos of running while held together and to my transom with rope after a cellulose-derived kinetic disassembly, was the same way, but had slightly different preferences.
Ergo, it may be more a matter of finding the motor's specific starting quirks when hot than an indication of a specific mechanical problem. That might potentially explain why I had no issue starting it hot on Thursday, too: I have spent a LOT of time over the last six months dialing in rope-start two strokes which you really want to get right in as few tries as possible. The default for me when hot is no choke/prime, but maybe 1/4 throttle.
I didn't do any engine work before sending her off (partly there was no time, partly the link and sync was fairly good right out of the gate). But all carbed engines can benefit from a careful eye.
The other think that springs to mind is that those engines had a solenoid for the priming system that pumped in the rich mix for cold starts...they also have a manual valve that can be used to block the fuel passage open.
It's a great setup if you're familiar with it: even if the solenoid fails, you can use the valve to get the rich starting mix. But if the valve is in the wrong position, you're getting a richer mix than necessary and it'll be hard to start hot. The starting was good enough when I used it that I didn't verify the position of the valve but it's dead simple to check; it's a little red switch on the left hand side of the motor, if you're standing in the boat facing the engine. It should ordinarily be pointing aft IIRC. I didn't think of it because I don't personally own one with that setup and haven't worked on one in a long time. But after thinking a while it popped into my head.
Here's a great video from Dangar Marine (who I love) that explains the system in some detail:
Anyway it's possible that the heat was getting to the ignition components; it got worked really hard for a long time. But I have to say that to me, the symptoms sound more like an over-rich mix for a hot motor.