Yes, this makes sense! How much water is in the bowl after the flush cycle and the tank stops filling? If the water level in the bowl goes down after a long period of time, you could have a crack in the trap that is letting the water in the trap to leak out. Once the level of water drops, the "seal" of the trap is no longer stopping the sewer gas from coming back through the toilet & into the room.
At this time, I would have the toilet re-installed on a new wax ring. During this process, verify that the flange is resting on top of the finished floor (proper height) & has no cracks or breaks, you may have to clean up the old wax for a good look. Sometimes the flange will end up with an almost invisible crack, that when the toilet is remounted, the tension of the bolts cause it to expand, thus letting gas escape into the room. As far as cleaning goes, it is possible some solid waste has gotten up under the rim, but this is rare and has a different smell compared to methane (sewer) gas. If no solution is forthcoming, a professional smoke test (performed by a plumber) of your sewer and vent system would give defined answers as to where the gas is coming from. Just curious, is your house foundation Pier&Beam or concrete slab?
The diagnosis of bad smells is without a doubt the hardest service call there is!
Your sewer pipes are not attached to your house. After long periods of time the house moves (settles) and plumbing gets loose, comes under stress, corrodes & generally starts wearing out from day one. I could write a book about the many causes of pipe & connection failures I've seen in 22 years on the job. Merry Christmas to you and yours as well & good luck.
P.S. have the wax seal replaced first, then if the problem persists, have the smoke test done.