(Please note that I'm not the original poster.)
Okay, I removed the tip from the solenoid plunger, then helped my brother reorganize the shed while my sister-in-law mowed their yard.
Then my brother used it to mow our mother's lawn.
It sounded great and didn't stall or stutter, so I'm convinced that the loose tip was the (current) problem.
After that, I took a suggestion from ILENGINE and tried a bit of hot-melt glue to reattach the tip to the solenoid plunger. I reinstalled it yesterday, so we'll see if it holds-up when I mow the lawn next. The glue that I used was just whatever residue was left in the gun from its last use. Hopefully, it won't remelt and clog-up the carburetor.
Thanks all for the help!