I would agree that the quality of the amplification and the choice of speaker are both variables here. In my experience, Klipsch has incredible presence but the sonic challenges you state are common and dissatisfactory. I’ve found Klipsch to work better with beefy SET amps or high current (not necessarily high power) solid state amplifiers. Both of the amps you mention aren’t the best for that.
I had the Prologue Premium about eight years ago. It was a great amp, but it wasn’t until I moved up the chain to realize how much more oomph a better amp would deliver and how much better a speaker could sound with it. If you plan to keep the Primaluna and haven’t upgraded from the stock tubes, I would highly recommend upgrading the output tubes to Gold Lion KT88 if you can find them (seems stock is low nationwide). They are amongst the beefiest sounding out there and will give you a considerable improvement in what you are looking for over the stock EL34.
i agree with others that power cables and supplies will also make an impact, but in my experience it affects the whole frequency spectrum and not just where you want to emphasize - so while this is worth looking into, it may not easily solve your problems.
If you upgrade your speakers to ones that have more midbass and midrange density, chances are they will also be harder to drive and require a more capable amp.
Speaker suggestions nonetheless, I would highly recommend looking into Legacy Audio. It’s just above your price range, but the Legacy Classic HD would be one I would recommend. You may even find one on the used market for much less and well within your budget, but ensure that it is the most current generation model as the Classic HD has been around for over a decade.
Ive also been very impressed with the Wharfedale Elysian 2, and it looks like there is a nice pair on one of the used marketplaces that I have no affiliation with. And don’t let the small form factor fool you - that speaker lets out a big sound that flows well and offers good clarity, all while delivering a bit of the Wharfedale/British hifi warmth.
Both are fairly easy to drive and would work well with your Primaluna, especially if you upgrade your output tubes, which I would recommend doing first.