According to George :
Stihl FS36 string trimmer. Walbro carb, diaphragm pump.

Sad story: starting last year, it would lose power after running for 15 mins or
so. I could usually keep it going by nursing the throttle for a while, after
which it would return to normal power for a few minutes, and then do they same
thing again. This year, it did that same thing the first time I ran it; but,
after that, it would die after running for about 30 seconds. I restarted OK,
but then just died again.

Your initial problems sounded like the one I had with my FS85.

Turned out to be that the spark arrester on the muffler was plugging
up. It would run somewhat weakly for a while, and then quit.

The small engines guy simply ripped it out, and it worked fine ever
since. [Spark arresters aren't particularly critical where we live.]

Now I'm wondering whether you jarred loose/plugged a fuel line.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.