Sorry, I’ve been out of town and driving/traveling a lot.
For your budget and requirements, the Krell Foundation 4K comes to mind. It’s $7500 retail and about $4500 used. Make sure to get the latest "4K" model. The earlier model does not support 4K. The Foundation has excellent power supply and audio stages - excellent audio resolution. It is only a 7.1 processor, so you are paying for probably the best 8 channel processor that still supports 4K (everything else is already doing 13-15 channels). Krell has it’s own ARES room correction, which is pretty good but it’s been said that others are better. That being said, the Krell does not support HDR (high dynamic range, which is extended color depth). If HDR is a requirement, you will need to look elsewhere.
If you want HDR, maybe look at a used McIntosh MX-160 (probably around $8-9k). It’s based on the Lyngdorf DSP/processing board and software, but uses all McIntosh power supply and DAC/analog stages. The configuration flexibility is excellent for a processor. The room correction system is supposed to be very excellent.
Alternatively, I have heard that the Audiocontrol M9 has excellent sound (retail $8900). However, the configuration flexibility is somewhat limited (i.e. you can only do a global subwoofer crossover frequency settting, etc.).
A NOTE: I am not a fan of room correction. I prefer doing acoustic treatments and very minor manual EQ if I can. However, others love room correction.