It's a common issue with H&K amps. The original design was aimed more at metal high gain playing than any other form and, while they have definitely evolved, they are still made by German engineers with German ears. There are certainly things you can do which will help but they take a little effort and research - and an open mind which can question and reject "common knowledge". (That term does not include "knowledge" which is not so common, it is just internet rumour and uninformed gossip which generally does not stand up to rigourous engineering testing).
The starting point is your speakers. All H&K amps show a marked sensitivity towards which speakers they are paired with. Search in the archives for any thread which is speaker related. You will find more under the GM36 than the TM20D which are more modern models. They all have a family similarity in their fussiness though the GM36 is more talked about than the others. Anything you find there will apply to your own amp too. I am a GM36 user who started with a TM36 and both of those models definitely do not like sharp speakers at all. They icepick with the wrong ones. You say "vintage speakers". If you mean Celestion V30s or any similar Celestion model I would say there is the start of your troubles. My GM36 and many others reported here hate V30s like poison. H&K speakers have V30s in them but they are specially tuned by Celestion for H&K which makes a big difference. I changed to G12M Greenbacks and things improved. I also have a set of the mythical Hendrix model G12Hs from the 1960s which sound good too, though not too much different to the G12M if I am honest. I would say trying out, actually auditioning with your TM20D, other more mellow speakers is a must first step.
Next, forget about spending a lot of time (and money) chasing the perfect valve!!! Don't waste your time up that cul de sac, they will make only a minor difference if any you can detect. There is a lot of hype and rubbish talked about how valves are massively different. It's tosh! The engineering is clear and easily demonstrated. Any differences are between individual valves themselves, which all have parameter spreads from new, and not between valves from different manufacturers. The average guitarist refuses to accept that the specifications laid down so a valve can be labelled a particular type number, 12AX7 for example, prevents anything significantly different in their action in a circuit. When you have a production tolerance of 20% in their parameters, a difference of even 1 or 2% between different manufacturers' production runs is totally inaudible. You can (and probably will) ignore this engineering advice in favour of the non-technical and unfounded accepted view of this, which is fed massively by those selling the valves to the extent of downright lying, but that is your prerogative of course.
A last bit of advice. Read this thread:
Understanding and using the H&K input buffer It is based on my GM36 as that and the TM18 are the only models I have full info on but it applies to all amps, even some non-H&K models, including your TM20D. The H&Ks all share a solid state input buffer, exactly the same in the TM18 and GM36 so presumably in the others too, which can be absolutely clean, mildly valvey, or a pseudo Tubescreamer depending on how hard you hit it. If you are using distortion or fuzz pedal(s) which distort and distort then output a high level of input to the amp's front end you are going to get a harsh base sound. You will need to understand how distortion is built up in the pedal + TM20D chain and be prepared to trim your sources of distortion. Read that and have a think about it.
A couple of other things to maybe try:
Stevie Ray Vaughan used to use his amps with thick blankets covering them. You can clearly see this in the picture I've added. It does reduce volume a little but can also take off some of the top end hardness. I have tried this and it works! It deadens the top end slightly and gives a more mellow sound. That suited me for most of my playing, especially at home, it may not suit you. But it is easy, cheap and takes only a minute to set up and makes no permanent alteration to your gear. You could also look into "Mitchell Donuts". This is a clever and cheap way to improve beaming in speakers which also mellows them a little. It takes a bit of setting up but it is interesting and it genuinely works to reduce beaming. Mitchell is an acknowledged leading expert in audio and acoustics who works from the engineering data produced in his own company laboratory and he has made the details of this free to anyone. There is a "Beam Blocker" device being sold which may help with your problem too. I warn you, remember what I said about "common knowledge", it won't and could never help with beaming despite the manufacturers claims! It may sound better to someone's ear but it isn't because it helps with beaming. The makers' theory of its action is based on a lot of absolutely incorrect assumptions which guitarists simply don't want to consider. You just can't fudge the Physics!
Lastly, a bit off the wall these, have you tried turning your speaker around and facing it backwards? The idea of bouncing the sound around the room can mellow it considerably as reflections absorb the top end more than that lower frequencies, as long as there are not too many smooth reflective surfaces about. And our founder and father figure here at the forum, HwyStar, always swore his Marshall cabinets sounded much better when he cut holes in the rear panels to open them up like most Fenders. (Or was that Namklak?) That is extreme but if you try the cab with the back removed and it helps you could make a new back panel for your cabinet from a scrap of plywood. I've done the opposite with some 1x12" cabs and made closed back replacements and that changed their character a lot.
It's not an impossible task to sort this out but you have to remember the advice, "in order to learn something new you have to be prepared to let go of something you learned before". The biggest problem in the search for an acceptable tone is a closed mind which believes what it has first been told without taking the time to investigate it thoroughly, then will not question it in the light of new information. And note, I deliberately did not say "perfect tone" which does not exist, even for an individual it changes day by day. "Tone junkies" are just that, junkies. Did you ever see a happy one?
Keep in touch and let us know what you think and what you find. It all helps others in their playing.
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