HP or Dell for computer listening ?

There will be no difference most likely. Using an off the shelf PC for music usually isn’t recommended because of the large amount of noise, electrical and otherwise, that can effect sound quality. But if you’re determined to buy a sub-$1000 PC, whatever one takes your fancy, whether it’s color or features or something else.

It's also quite possible that the Burson will be insensitive to or more likely lack the resolution to be able to playback the PC noise anyway.

I mostly listen to youtube, sound quality is never great there and can at times be just terrible. I have never used outboard dac/headphone with computers before, but now that I got much better headphones, I would like to improve it further, that's why I bought the Burson. Don't have it yet. I am thinking desktop computer.
Everything will be plugged in Furman power strip/conditioner and I will be using Wywires Silver USB cable.

Either of those computer brands would work, shouldn't be any difference in this application.

Regarding reliability alone, in the last decade as an IT exec I have sourced and supported probably around 5k laptops, all either Dell or HP.

I give Dell the edge for reliability and serviceability, especially right out of the box. An unnatural number of HP machines arrived DOA.

In the sub-$1k category it's more a horse-apiece. Just stay away from refurbs and get the service plan.