I assume you've used the Avermedia in Linux before? In my searches for capture hardware they weren't even on the list of Linux supported hardware, though that may have changed, it has been 3 years or so since I last looked.
But as for what level of hardware you need to stream from a dedicated box, I would say either an i5 or Ryzen 5 would be plenty of CPU horsepower. 8GB of RAM seems to be the minimum recommended for any build these days, and that's plenty for streaming, even if you're dragging in browser-based sources such as Twitch chat.
As for GPU, I don't think you need one, unless the CPU you choose doesn't have integrated video. OBS is fairly light on graphics horsepower when it's not pulling game footage from the GPU. I know some people would disagree on this point, but they are generally in the category of people who don't get down in the weeds of where performance actually counts. I've found on almost any graphics card, OBS tends to use a higher percentage of PCIe bandwidth than it does actual GPU time, and that's entirely due to the capturing.