[Arran]

Speaking of shotgunning components, I've noticed a tendency to condemn all of the tubes in a device as the cause of a fault, in my experience tubes are seldom the failure, when I get a new set I check the tubes out of habit but have seldom found any that would prevent the radio from playing. In a series string set I will find ones with an open heater, usually the power output tube since they have the largest heaters, but not usually in a transformer set. One thing I have noticed is that certain tubes will wear out faster then others, depending on what they do, the mixer-oscilator/convertor tubes are one, first audio/second detectors are another, one that seems to last and last is an IF amplifier tube. In this set they used a 6H6 as a second detector/AVC tube, and a 6F5 as a 1st audio tube, and the 6H6 is another tube that seldom wears out. To be clear unless a tube has very poor emission, such as under 40% the set should still play, it just won't perform as well, obviously tubes with shorts in them, or open filaments will stop the set from playing at all.
The 6K3 is another model pulling what I call the "tube stuffing" scam, they added a 6F5 and a 6H6 where they could have used a 6Q7 or a 6R7 to do the job of both tubes, this was all too common in sets made around that time, and if they added a magic eye it was then a seven tube model, but in reality you were getting the equivalent of a five tube set. Instead of adding that tube they could have added an untuned RF amp, or an extra IF amp, with a minimum of extra parts, but I really think that the 6H6 was one of those tubes that was the answer to a question nobody was asking.
Also when I recap a set, I usually work my way through from one tube socket, or terminal strip, to the next, checking resistors, and replacing ratty wires as needed, it's much easier to keep track of the work I have been doing. In many AC/DC radios they have the filter caps mounted in a cardboard tube under the chassis, and I will often remove that first, and replace it last, as it's often in the way of accessing other components. I like to restuff the tube as it makes for a cleaner installation then trying to mount the caps under the tube sockets.
Regards
Arran