Can a marginal transformer be responsible for random standing pilot light cutoff?
Usually the pilot light is normally only controlled by the thermocouple/ sensor on systems of this age probably only a millivolt controlled gas valve.
What I have seen quite often is the burners get dirty some rust on the cast iron burner holes and even spider webs on them can stop the flow then a bit of gas is not burning properly and when the call for heat is gone the main valve shuts off (not the pilot) and then there is a puff of gas that flashes and that pressure blows out the pilot on shutdown.
I usually pull the burner elements and clean the tops with a wire brush , then clean each opening with a wire just small enough to go through (a small drill bit also works if there is a lot of rust. Don’t cut any more metal away.
After completing each element I reinstall them light the pilot if it went out and adjust the flame. Each burner usually has a opening that is adjustable on the back I take them all the way to blue and work to a small orange tip, then I start over, sometimes the adjustments on the first ones will be off. If they look good I will cycle the furnace a few times to see if it is going to blow itself out again.
There can be other things but with what you explain I would go there (clean) first.
As far as the age I have worked on “ancient furnaces” some solid cast heat heat exchangers (from the 30’s) that will probably last another 50 years one had all the replaced motors listed on the side and the 1 gas valve that was replaced because they wanted to add a thermocouple safety.
If the heat exchanger is in good shape you loose a little efficiency but a higher efficiency furnace will never pay for itself compared to running some of the old beasts that I have found that will probably continue with a new motor every every decade or two, check out the burners most come out in sections when cool lift pull away from the manifold (sometimes if you hit a stop rotate~90 while pulling to the back, once they clear the manifold pull it them out to be cleaned and reinstall in reverse order.