I remember another legendary TV story of someone offering to improve the alignment on an old vidicon tube camera. After many hours of adjustment he pretty much got it back to where he started. Ah, the confidence of youth.
I started doing electronics repair work in a TV shop in the early 60's. When PC boards began to show up in consumer products the problem of cracked solder joints at the lead holes became a problem for hot or heavy components having leads that passed through the board without being bent over on the bottom. This was a particular problem for resistors over 1/2 watt and tube sockets (if you missed out on tubes in school, look for vacuum tube in Wikipedia). The problem was caused by the repeated movement or thermal expansion of the lead at the soldered hole which eventually caused a crack in the solder or the trace at the edge of the solder. The problem was a great money maker for repair guys because around 70% of TVs with a given chassis would have the same failure at the same components. The bad joint could usually be spotted just by wiggling each suspect component in a dark room while watching for small sparks where the lead was soldered to the board. Once the first new TV model's problem was located, the next ones with the same chassis were easy to find. That thermal cycling problem continued for years, although not to the same extent when manufacturers learned to bend the bottom of the leads over before soldering. Finally component heat dissipation was reduced to the point where it would not cause significant thermal expansion of the joints. When I started doing equipment design in the 70's I always specified that any heavy or high wattage component would be mounted with the leads in an open rivet which was crimped in a plated hole in the board, followed by wave soldering the rivet and lead. I never had any reliability complaints about that mounting method.
I've seen canned "freeze mist" with a straw delivery attachment that allowed you to chill one component at a time, and heat guns with nozzels that produced a pencil-sized stream of hot air, so it is possible to be very selective with your heating/cooling for a short time (conduction will spread things around pretty quickly). Given the crudeness of the tools available, I am very impressed with the systematic approach and gradual narrowing of the target area. But the real key was noticing the "works for awhile" symptom and correctly inferring a thermal problem. Well done!