It’s actually a very good idea. I’ve seen a similar request on the forum here from a guy in Russia who wanted to check of the stove of his dad or mom was turned on for x-amount of period. They lived like a thousand miles apart, so … you can imagine the use case for something like this, I sure can.
I think there is a big IF (and not in a programmers way). If something went wrong your insurance might go nuts on this with all the hassle belonging to it. There are certain rules and regulations for gas and such and I for one, don’t like to fiddle with those. So there is another aspect here, not only technical (which would be very easy probably), but also some juridical stuff.
Sensing stuff as you described would probably be a good idea though. There are probably sensors (like ones for the BBQ) who can measure a good high temperature. Combined with the PIR it could be a match made in heaven.
The reason those safety stoves are so expensive is, I think, the fact that they are approved by insurers and such.