Seeking ideas about how to improve my system

@jeankunin, I know what you are experiencing.  At one point I heard a $2K tube system that I found more listenable and accurate that the much higher priced system I had at home.  I had long list of different components I kept switching out until I found an integrated with a built-in DAC/streamer having a sound signature I liked and I couldn't be happier.  I will end up going back to separates at some point again but at least I know that the sound I am looking for exists and which brands of amplifiers and cables it took to get there.

While room treatment should make a good impact on sound, I have found that you can at least tell if you like the overall instrument realism, timbre, etc. straight out the speakers.  You can then use the room treatments to fine tune the sound to balance highs, mids, and bass if there are issues.  Of course this could be just me. :)

I think you could gain a lot by better speaker placement. A 20'x20' room is quite susceptible to standing waves.  The best speaker and listening position placement will avoid the positions where there are peaks and nulls of the standing waves. This will keep the speakers from unnecessarily exciting the room modes. Assuming your room is indeed 20' deep exactly, the major positions to avoid are 20'/2, 20'/3, and 20'/4, i.e. 10', 6'8", and 5' from the walls. A good place to begin would be to place the front of your speakers at the midpoint between 5' and 6'8" from the wall behind them, i.e. 5'10", and at the same distance from the side walls. This will make them 8'4" apart on centers. The listening position can be placed similarly from the wall behind it.  If your room is not exactly 20'x20' adjust the numbers appropriately. Fine tuning must be done by ear but this will give you a starting point. You might find moving the listening position a little farther from the speakers helps them blend a little better.

Well, you are in trouble, in financial trouble. Room treatment comes last, if your near field listening experience confirms your impressions.
You may need to spend ten times more on your speaker based system to equal headphone based system. Hopefully less.
Speakers and speaker cables is where I would start. I would get rid of that Nordost interconnect as well, regardless of where it is now.
I wouldn't upgrade analog front end until I get digital sounding about equal to digital through the headphones. Then I would take care of analog. Good luck, this is going to be interesting.
Yes, my speaker based system is about ten times more expensive than headphone based system. It sounds better but the difference is moderate not big, except bass which is much better with speakers.

I was an early responder to this topic string, who mentioned I was in the same place as I had upgraded my headphone system to be much better than my main system. After upgrading my main system I haven’t picked up my headphones. I have a little time now to think about your system. I looked up the equipment you have that I didn’t know. Interesting. I would say given what I know about your equipment you have it pretty well balanced at this point. It looks to me you put a lot of thought into it. I don’t know much about the integrated amp and have to think that this is one of the opportunities. A really good preamp is a critical asset.
.
My rule one is: minimum upgrade is 2x cost of each component. This will give you a "wow" that sounds a lot better. High end audio has been my passion for over 50 years, so I have gone through many upgrade cycles to reach a new plateau each time. Then stay there for 5 to 10 years.
.
First thing. Given your equipment, your analog end should sound better than your digital end... at least the same. I know VPI... so I am thinking you should consider upgrading your cartridge. I would double the cost. Still not an astronomical expense... I just can’t see anything else holding it back.
.
Overall concentrating on the following will improve both analog and digital ends. So, you get more bang for the buck as well.
.
I have effectively used very inexpensive speakers with subwoofers for my system for years. But ultimately they can hold a system back. I think it likely the SF at that level may be a bit of a weak link. Sonus Faber makes great speakers... I endorse them Olympica and above. But some of the less expensive are light on detail and overall presentation. I had some Cremona... I liked the sound but they were not very detailed. A word of caution... I know you want more detail... but be careful not to step too far and loose the musicality. For a long term plan... I would upgrade the speakers first. Think about going to the $10 - $12K range. This would be a project in itself. Then preamp, and the amp. While I don’t know much about that integrated... so much of your equipment seems good it is hard not to suspect it. Separates at say $5K or so each, but consider more on the preamp... it sets the stage for everything. At the $5K component level you are on a whole new plane... and with speakers in the $10K range... it should be really spectacular upgrade. The experience should best your headphones, maybe not completely in detail but in overall presentation, imaging, bass, musicality.
.
I suggest you audition a lot of speakers. Normally I would recommend Sonus Faber speakers given the music you like... but perhaps you want more detailed oriented speakers. Wilson maybe... or maybe check out SF Olympica if you like the overall tonal balance and musicality. They can do a lot with really good input. You could go planar... but it opens up such a can of worms. I pursued planar for most of my audiophile years and ultimately decided the incredible fine detail was really secondary to musicality, and now have an all tube system... has the detail, but doesn’t stick it in your face... it has fantastic rhythm and pace, musicality, bass, and detail.

Fiddle with your room and speakers in the mean time. A lot of improvements can be made there. I am lucking to have accidentally bought a house with a fantastic huge open plan man cave with near perfect acoustics... then I added diffusers and tube traps... I was more lucky than good.
.
Yeah, I think my main system is about 10x the cost of my headphone system... but then it blows away my headphones. Maybe equal in detail, but 10X better overall. An absolutely worthwhile investment.

My point OP, simple is better..:-) Especially with vinyl..

Regards