Warm vs Revealing—the struggle for balance

Yes... that's a common problem that many struggle with.  In my experience, the best solution is to own tube gear that allows for a variety of different tubes to be used to fine tune the sound to your liking.   While tube rolling can be expensive, it's less expensive and stressful than having to swap components... and I find the experience of tube rolling more rewarding and educational.

It sounds like your problem might be bright gear not revealing gear. A system can be very revealing and not be fatiguing.

I’d start with a pair of speakers that you really like. Then get an amp which pairs well with those speakers, and so on. This may cost more and take more time to put together your system, but it’s better than constant frustration.

If you live in an area with an audio dealer or dealers, it may be cheaper and easier in the long run to work with a dealer, buy new and get a system that really floats your boat. Buying new from a dealer does not guarantee that, though.  If you go the dealer route, take your time, make sure you trust that he's got your listening enjoyment in mind and not just a sale.  He should allow you to try the gear at home before the purchase

Amps that can run multiple types of output tubes are, by design, compromises.  They don't optimize signal transfer of any output device,  so all possible output devices are not optimized with respect to signal transfer in their operation.  For some this is OK, even desirable, for others it's not either.  For those in the latter category, the sane course od action is choosing speakers based on their signal transfer and then having amplifier(s) built that optimize the output devices' transparency and are then tuned for the desired degree of warmth (i.e. distortion, which should be minimal if amplifier + output devices are well optimized) by choice of  coupling capacitors...in this way both revealing and warm can be had

I think, you are headed in the wrong direction. You don't really want to find warmth/detail balance, you want to find the sound that is as close to the reality as possible. Then you fine tune, including with tubes. 

I agree with tomcy6,  get the speakers that you like a lot and bring the best out of them. It will not be inexpensive.

For example, I currently use 4 different pairs of stereo speakers, several amps, sources (essentially multiple rigs), a multichannel rig, etc in 2 rooms. It covers the different flavors, signatures, etc with different types of music that I enjoy (while my moods fluctuate).

1 in 1 out when it comes to equipment/rigs is not a good idea at all (as you're noticing)....

Apparently,  "it is a common problem that many struggle with", as indicated by one commenter. So, let us probe a bit deeper into the psychological make up of many typical dudes. Stop telling yourself that the one sht you got is the greatest sht on earth somehow (because you paid for it and you are somehow the most intelligent dude that ever lived n all). Admit to yourself/acknowledge the strengths and weaknesses of the sht you own (weaknesses exist in all kinds of cost no object gear too). Don't be a feral fan boy of one brand, etc i.e don't put all your eggs in 1 basket. Keep an open mind and keep some options open.