New DAC, what should I look for?

Since you have HQ Player, I would suggest you look for a used T+A DAC 200. It has separate processing paths for DSD and PCM and does spectacularly well with high res feeds from HQ player. Yes, I own T+A gear, but this recommendation comes with a lot of outside support. There are forums on audiophile style dedicated to the unit and HQ player you might want to peruse. 

The DAC checks the boxes you list regardless of format (but is shockingly good on native DSD upsampled from HQP) and is within your budget pre-owned. At least worth exploring.

Best, 

The sound profile that I'm looking for is smooth and spacious

The PSA DirectStream Mk1 isn't smooth or spacious? Just want to clarify.

I love my Holo Audio Spring 3. Very smooth, spacious, good detail, solid all around. I will never look for another DAC. This part of my system is done.

Here are a few worth looking at given what you’re looking for.  Best of luck.

https://www.mojo-audio.com/demo-mystique-y-am-d-a-converter/

https://www.audiogon.com/listings/lisbcf2a-denafrips-terminator-dac-15-terminator-new-terminator-15-dac-plus-da-converters

https://www.lineartubeaudio.com/products/aero-dac-digital-analog-converter

OP:  The sound profile that I’m looking for is smooth and spacious.

Have you seriously tweaked the upstream components first, the modem & router?  With: LPS’s - digital filters - quality ethernet cabling - power conditioning?   

Agree with a Holo Audio DAC to match your Red streamer.  I love the sound of my Cyan 2.  Their May DAC is top of the line and you should be able to find a used one for under 5K.