It’s been decades since I had to mess with a faulty stove so forgive any memory lapse, but with our old Magic Chef on the Pace Arrow the behavior was occasionally similar. It was always a dirty valve. That’s not to say it couldn’t be your thermocouple, but I used to disassemble and clean control valves to get mine going, or reset or replace O rings to correct leaks.
Make sure the thermocouple isn’t fouled or misaligned... such is known to keep a fridge burner from staying lit. Also see that it’s wire is solidly connected at both ends.
It’s unlikely you need a new cooktop, Dan, but maybe try to locate a repair kit if cleaning or adjusting things doesn’t do the trick.
Joel