Best set-up under $20K

Grover30,

Did you play with the treble switches before chasing amps? But I can see class D being a no go on these (I use a MC462).

I run subs with my 4367 (but I do with all my speakers) and I have the UHF at -0.5db  I also listen to hard rock/metal (pop and audiophile stuff too). I think what makes them so good with old average recorded music is the great dynamics. Even 80s music with poor dynamics seem to find dynamics when played through the 4367.

The speaker is so detailed in the upper frequencies it does almost sound bright. But in direct comparison with the Revel 228be the 4367 was a touch more down in the treble which surprised me. The 590s, (my surrounds) are much brighter.  

I had an 802....quickly tired of them

By the way - I have the Hegel H590 integated amp.  I got it to drive my Harbeth 40.2’s because Harbeth uses the Hegel amps to demo their speakers at shows.  It does a great job at controlling the Harbeths, which many amps struggle with - it adds clarity and resolution.  
 

You can bypass the internal DAC to use an external DAC.  I use the Schitt Yggdasil , which is superb.  I assume you can do the same with the H360.  Which is why I suggested to stay with it for now.  You can upgrade later if you need to.

Do yourself a favor, and don’t even think about it, seek out a pair of pre-owned Revel Salon 2 speakers, and thank me later. Revel caught lighting-in-a-bottle years ago when they first introduced the Revel Salon 2s, and it took years for the other brands just to catch up. The Salon 2s will amaze you.  So, no need to be concerned about the age of their technology. The Salon 2s are a true end-game caliber speaker, and still one of the best values on the planet!!! Now, Salon 2s need lots of power and current, but not necessarily insane power, in order to drive them properly. I would say, at least 300wpc into 8 ohms will do the trick. Start there, and figure out the rest of the system with as high a quality cabling and other components that you can afford, and you’ve got yourself a world-class sounding rig!!!

I love all these amp and speaker recommendations.  It's like recommending a paint job on a ship with a big crack in the hull.  

+100 on the posters suggesting the digital chain drastically needs an upgrade before any other gear should be considered.  Airplay is far from optimized.  A good wired streamer and DAC is absolutely the next step.  It would be challenging to determine what else to do until the source is better optimized. 

Tidal is supposed to be rolling out non-MQA high resolution and so make sure your new digital rig is capable.