Gentlemen,
Thank you so much for your help!
After removing the grub made the engine stop running, the fact that I know compression is perfect hot or cold, the fire is strong, hot or cold, all your suggestions about it being fuel related started making sense.
When I started digging into this saw, I realized that I gave bad information at the beginning of this thread. I had NOT rebuilt this 028 carburetor, but it was the 025 that had been rebuilt. I had mentally ruled out carb issues because it runs so flawlessly when it runs. So I decided to immediately rebuild the carb. While shopping for a rebuild kit, I saw a replacement carb online for $16, so what the heck, I ordered it, too. Worst case, I have a backup.
When doing the swap, I noticed that the pulse line was only pulled about halfway onto the barb. Interesting! I also went back to the original coil and I replaced the grub line.
Once back together, I pulled the cord. Started instantly and ran well, as usual. But this time, she kept running smoothly for 2 hours of hard use.
She is fixed. It was definitely fuel related. I'm 50/50 between suspecting the pulse line and where the carb connected to the intake. The new carb came with a gasket for that place, and the original one didn't, so I'm wondering if that wasn't opening up when the saw got hot.
Half of me wants to install the old carb with the new gaskets and see what happens, but right now, I have a lot of wood to cut before worrying about that.
Again, thank you for all your help, and sorry for the inaccurate information.