I'm with @russ69  on this. You have a mixed starter system now. The mix is the Yammy HT receiver in with 2 channel.. I like the Yammy's quality and sound And they make decent integrated also which is what you need . There are different priorities between the 2 channel and HT.  BTW, A 2 channel HT is fine for me and many others who have used it. .If you are adventurous a tube integrated may sound better to you also. But expect a little maintenance IE tube changing when bad. But many will last 5-10 yrs so its not an extraordinary amount to maintain. Then after the amp, the turntable needs upgrade. Then the phono preamp. Cartridge is alright for the time,have fun

Upgrade path question with $2k to spend

I really don't see how my $2,000 Yamaha running in pure direct mode with 200wpc is affecting the sound quality.

The question then is this, Why do you want to spend $2k  to upgrade a sound system if all sound is equal? Do you really think the priorities for  movie dialog with a few explosions and music which covers 20-20Khz signal are the same? if you want HT then the Yammy is fine. But I've been down that route and it is not the path for good 2 channel. But its your $$ to do as you please

Ok so can I run both a 2 channel integrated Amp AND my HT Amp to my speakers at the same time and just switch off somehow?

That is possible but somewhat impractical. IF you have banana plugs on your speaker cables, that would be pretty simple to switch out. A speaker switch is possible but it also can cause a loss of fidelity. The sub may be an issue. I can't say for sure. But what I will say is that many of us tried the 2 channel & HT all in one system. I even had a 3 channel amp/processor tied into the 2 channel stereo once.  IMO it was one GIANT cluster **** with very little return. I now run the TV through my 2 channel. Sounds great, especially for concerts. And it works fine for HT as well. Many have had to decide this issue of which is most important, HT or music. But hey, I understand the frustration involved