Upgrade path question with $2k to spend

Probably the OP doesn’t have room or budget, but my advice would be to keep the Home theater gear as a separate system, and add a dedicated 2 Chanel rig. They can share the same subs in some cases. I run 2 separate system in one room. The Home theater speakers are stuck in corners, etc as it’s not my priority and the 2 Chanel speakers are placed further into the room. Since I have a tube preamp and tube input section on my amps, I don’t want to run tubes while watching movies.

Definitely, definitely take out the Yamaha.  Doesn't matter how expensive it is or good it is for multichannel listening.  You want a 2 channel amp with some toroids.  Either an integrated or a  preamp/amp combo which has home theater bypass.  Then you can still use the whole system.  At $2000 budget, look used.

I’d not spend $2k for a phono stage. Heck you can find the PS Audio phono stage used for less than $500, which provides excellent results. 
 

i’d upgrade the Turntable. $2k goes along way on the used market. I just bought a new MMF9.3 for $1800. But no doubt, save funds for an integrated amplifier! Immediately. 

Replace Yamaha with Parasound Hint 6 int. amp.

I, too, have a blended system, but my AVR is an older Denon AVR-X4000 (which was their flagship AVR at the time).  The important thing is that your Yamaha appears to have pre-outs.  I run my Denon pre-outs through a Parasound P5 preamp then to a B&K EX-442 Sonata amplifier to power my Maggie MMG mains.  My surround speakers (Elac Debut 2.0 B6.2) come off the P5 and run through an Adcom GFA 6000 5-channel amp (nothing else is connected to it but those two surround speakers).  I don't use the internal Denon amps for ANYTHING - I don't run a center channel speaker.  Denons run famously hot already, and mine still does, even with all the power handling offloaded to two separate amps.  If you don't need surround speakers, great.  You probably don't even need a separate preamp, but I wanted one strictly for two channel music (or 2.2, as it currently stands).  I'd say that, from my angle, one of the easiest ways to improve your sound is to get your mains off the Yamaha AVR receiver and feed them from a separate amplifier.  That would be my first move.  If you don't want to add another component (which certainly makes sense ) then the alternative is to do as suggested:  get an integrated with a HT bypass.  The key to these blended systems (and getting them to sound their best for their respective applications) is to have a preamp somewhere in the chain with a HT bypass.  Gotta have it, IMO, to carve out as much of the "video" signal path from the music signal path.

Luck.