Tube amps - what 3 things…

Today afternoon I was spinning "Get On Board", Taj Mahal and Ry Cooder. Can't get enough of this album, but "Packing Up Ready To Go"... that song is something else. They sure found their sweet groove. Should you not feel the presence of that bass drum in your room, change your system. 
Oh, yes: recording is great, mastering is very, very good - and appropriate. It is all captured and preserved, from sweet rawness that usually get conjured during casual jam in a forgotten roadhouse, to the reverberation of the room they recorded it in. Absolute treat for music lover and audiophile.

My 3 most important things - for my system, not only for an amp:
1. True to the recording - reproduce what is recorded, no more, no less. If there are "spatial cues", I want to "see" the room. 
2. As close as possible to the real voice(s) and instrument(s) - reference is the live event and, in my case, years of being musician
3. Goosebumps. A lump in my throat. On some recordings happens. On most - it does not. I need to be taken, transported to the event, even for just a moment. System is the vehicle. Music is the fuel. Rare are the pieces do have such a beauty that can transcend poor reproduction. Conversely, if music is crap, no system can make you suddenly hear The Masterpiece. 
In too many years I'd care to mention I had many different systems. In early eighties, I had one ghetto blaster in my apartment. I even tried D class Bel Canto with Dynaudio speakers. Now, I am on my second Audiomat Prelude Reference. I sold the first one, to "try something different"... I also have Audiomat Phono 1.6 - which would classify as SS. My MD806t uses two 6992. Speakers are Capriccio Continuo Auralea 309, very good match with the amp, and the room. 
I prefer good sound, and don't care is it tube, SS, D class... nor I care about features. Less is usually more: I prefer integrated with passive preamp - power amp with input selector and volume control. 
I heard two different Zu speakers, one with Decware EL84 and another one with Trafomatic. If I would have your dilemma, I would try to hear those manufacturers. Trafomatic was more to my taste. 
Personally, the only integrated amp, besides Audiomat, that I would buy is Ayon.

Also, tube is just a part of the design. I heard EL84 design that was well balanced, with a nice and tight LF output. Also heard 6L6 design with mid-bass hump, and quite flabby down low. Good mids, but nothing else. So, which tube should be followed by which iron, which caps etc. Avoid all of that and go out and listen. Ask Zu people. Visit dealers. We can all give you great insights, but ultimately, those are only our, ultimately subjective observations. Hearing is very subjective, just like any other sense: taste, touch, even vision - just read the studies on crime eyewitness reliability. 

I agree with the previous comment on transformers being the most important component.

I recently posted about some coupling capacitor upgrades on some chifi integrated tube amps and it made a big different in sound quality.

But just recently I gave some Icon Audio equipment a try again (had one several years ago). In direct comparison to the chifi gear the resolution doubled. I am hearing elements of the music at are either veiled or completely missing on the chifi gear even after upgrades of tubes and capacitors. This is primarily due to the better transformers on the Icon and of course the actual circuit design.

Next most important thing is do not listen to Youtube reviews. I think many of those guys are being paid off to promote average equipment or they are half deaf and have not experienced truly good equipment.

Once you find a good piece of equipment you can tweak the tone with some nice tubes or upgrading a few coupling capacitors. But it would be unrealistic to try to upgrade transformers.

I agree with the previous comment on transformers being the most important component.

Until they aren't. Tube amplifiers with output transformers, regardless of how good they are wound are not as transparent as their OTL or direct drive counterparts (given a level playing field). I've have all three design types and it's pretty clear what an output transformer adds to the mix. Getting rid of the capacitor at the output is also a benefit.

I have a selection of amps including Naim (solid state), Vincent pre / power (Hybrid) and recent Cayin (tubes). Currently despite being the cheapest the Cayin clearly wins in my system (but with EL34 tubes not KT88's). Three lessons, 1, forget the technology listen to the amplifier. 2, some speakers suit some amplifiers, some do not, listen to speakers first then buy an amplifier that suits them. 3, don't be hung up on price, I auditioned at least 20 tube amplifiers before buying I am perfectly happy with the cheapest, by a long, long way. Features, don't pay for ones you don't need, they add cost and complication. Think cars with multiple 'drive' settings, the 1st month you try them all, 95% will never change them after that. 

My Firebottle is from late 2016 (it has its birthday on the bottom with Had's signature) I and  bought it in early 2017 for 1100 bucks or so. I'll never sell it.