I think you are off the mark with your analysis of the problems. Forget the caps and opamps! They would be a secondary problem. You have a working amp so the caps are at least good enough to get it there. Worry about that later. Practically, opamps do not suffer from a fading of performance, they either work or they don't. Again, they are working at least well enough to run the amp. Worry about that later. And there are NO magic opamps in our line of work. Despite what the hacks butchering pedals may say, the truth about opamps is that their effectiveness does vary massively, just NOT in audio work like this as long as the original design work is carried out correctly. Opamps give poor results only when they are driven outside their range of characteristics. That means that as long as you make sure to take care of a number of factors such as, their drive and load impedances stay within the correct limits, they are operated within their frequency range, and you never allow them to clip internally (there are excellent simple ways of preventing this which are adopted in a few good quality amps but ignored in the vast majority) you cannot tell the difference between even the humble TL07x series and more exotic ultra expensive types. For the circuit designer the array of types is there to select the most suitable when the parameters like low impedance drive and high frequency response or low offset requirements need to be met. The H&K designers have done their work well. They have designed good opamp stages and the opamps they use are working well within their capabilities. You will gain nothing spending time swapping them out then trying to convince yourself you can hear a difference. It can't be measured so it can't be heard unless there is a blue moon and a unicorn sprinkled pixie dust into your ears, and I can point you at a high level paper, (a long paper which critically compares many opamp types right across the range), where this fact has been put to the test and found to be true.
For the others out there who disagree with this I would suggest you come in on the discussion and ask for the evidence before you make the usual "utter bollocks, everybody KNOWS that they make a difference" argument. You can then make a fool of me by showing me where that evidence is wrong or perhaps point me to other evidence which shows, categorically and factually, that there is a difference in a well designed circuit. Most pedals and many amps are NOT well designed circuits!!!
So to the TSC. It works by measuring the voltage across a small (1.5R) resistor in the cathode of each power valve. That voltage is a direct measurement of the cathode current in each valve. Also in the cathode circuits above each resistor is a MOSFET. The TSC processor is monitoring the voltages across the resistors individually and controlling the gate voltage of the two pairs of MOSFETs on each side of the push pull output circuit to set them to a preset level of bias current programmed into the TSC system. That's why there is no bias adjustment in the H&K amps. It maintains a good overall balance between the two sides of the output stage. This has to take place intelligently to ignore the part of the cycle where there is massive output current, hence the processor controlling it.
Some solder faults which increase the resistance in this area would have some effect. If the 1.5R resistance increases due to its joints then the TSC will be thinking that the current is higher than it really is and it will overcompensate to bring it down. It would settle at a lower current than it thinks is present. It is possible that bad joints in that section could be at fault and, as you say your soldering skills are good, it wouldn't hurt to correct them. I would say that it wouldn't be a good thing to just go over the whole board as most people advise. "If it 'aint broke don't fix it" is a good maxim to follow. It's much better to actually diagnose what the fault REALLY is than to just assume you might fix it by doing a few blanket things and then hoping you got it when the fault goes away. I've seen faults come back a short time later too many times with that approach. Remember the days when we "fixed" our televisions by knowing the right place on the side to give them a whack with the flat of our hands? That ALWAYS "fixed" the problem didn't it? (For a short while anyway.
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Look at your TSC readings. You said you got (13, 26, 13, 12) when you first checked. Then you swapped the outer and inner pairs and got the same result. Then you left it overnight and retested and got (21, 13, 12, 13). That's good!!! Compare the two sets of readings. Have you not spotted that you swapped the right hand pair and the left hand pair over and the results went with each valve exactly - eventually. You seem to have one unstable valve which reads 26 and 21 but all of the others have behaved exactly as they should. The delay in testing may be important. It may take time for the TSC to recalibrate itself to the new valves but it eventually seems to settle down to what it should be reading.
Try swapping the left hand pair with the right hand pair keeping the inners to the inner and vice versa. I'll describe that a different way to be sure. Swap the inner valve pair with each other and then the outer valve pair with each other. You still have the same pairs working together but they have been changed over in a legitimate way to something you have not tried. Now test the TSC reading a couple of more times with switching the amp off and on each time. If it makes no difference then wait overnight and retest. As I said, it looks like the TSC takes time to recalibrate itself which is not surprising. Assuming you still have the valves in position from your (21, 13, 12, 13) test, after swapping 1 with 4 and 2 with 3, I would expect you to get (13, 12, 13, 21-26). If you get pretty close to that then you have no problem with your TSC.
My advice is to fix that PCB mounting problem first, it is obviously badly wrong so there is a serious fault in the mounting. That should be an easy fix. You should find the boards connect together via small jumper cables between white sockets. You can't get them wrong as they have different numbers of terminals and they run pretty much close to each other. They should just gently disconnect without too much serious pulling and pushing. And no, these are not cheap connectors despite what they look like, they are good quality industry standard! Don't worry about them. The H&K amps have drain resistors across the power supply caps so, measure the voltage across them before you go in and you can be sure the voltage will stay down and be safe to work on. It should take only perhaps 30 seconds to come down to safe levels once the amp is switched off (and unplugged of course).
It takes a lot of work to get yourself into the mindset of only fixing what you know is broken and trusting what isn't. As a budding tech you need to come to terms with that, one way or the other. Are you going to fix amps according to what is broken and really needs fixed or just go by what every "guru" tells you you should be doing without any supporting proof that it has any validity? If you like the sound of the amp when it is fully working, there are a lot of reviews to show you how that should sound, then why assume that esoteric "common knowledge" weaknesses just MIGHT be there and that you can improve on the H&K design team by swapping out their faults? Remember that "common knowledge" is just that. It's knowledge common to everyone out there who knows absolutely sweet FA about amp design, components, and what makes an amp sound good or not.
EDIT: And, for what my personal opinion is worth, an electronics degree DOES mean shit! You've started your career the right way, with a good sound foundation of engineering knowledge and principles and, presumably, training in how engineering is at the core of amp design and function. Don't swap that logical critical approach for a reliance on what untrained people on the internet who don't really understand in depth tell you is truth. ALWAYS check out any claim for yourself and be critical of it. (My own included!)