I’m not sure of all aspects of your question. FWIW, I ran core on a Synology 1513+, which is a pretty meager processor, an Intel Atom D2700, with 4GB ram. It ran DSD256 fine, but choked on DSD512. Roon responsiveness was a bit slow. This was over ethernet. I don’t connect via USB and don’t know how that would affect things.
Now have a SonicTransporter i7 that handles it all speedily.
I have a QNAP TVS-872XT on the way. It is the sweet spot in their line for power to cost, though may still be above many budgets. It is Intel i5. I think anything from my 1513+ to the QNAP with i5 would run DSD256, based on my experience. Just depends how much power you want for general responsiveness. If you are going to use much DSP, I would go for the higher end of the range as well.
Then there is connectivity. 10GbE or Thunderbolt to get fast transfers if you do much work on the NAS from your computer or transfer files back and forth a lot.
As an aside, I will be putting the Synology 1513+ with at least two 4TB WD Red drives, maybe three, on ebay in a couple of weeks after I finish migration to the QNAP. An inexpensive way to get into, what is now, an entry level NAS.