As far as I’m aware, neither AMD nor Intel have such limits in their hardware encoders or GPU drivers. The only limit is performance of their hardware but 1080p doesn’t have too many pixels, pretty sure they both can handle at least a dozen of streams.
You’re free to use the CPU to transcode but you may only get a couple of streams. The quicksync on chip can do 15+ on the lower end but sacrifices encoding quality for speed. Though recent quick sync and nvenc quality has significantly improved.
Edit: Intel has added the dedicated encoder chip inside most consumer (almost no xeons) where as far as I know AMD is not doing the same thing (not counting GPU encoding).
Nvidia's H.265 encoder in Turing (the 2000-series of GPUs, many RTX branded) delivers superb quality for >=1080p30 real-time encoding compared to the alternatives (x264, QuickSync, AMD GPU) (x265 isn't really suitable for realtime-encoding).
The only reason I haven't yet planned a concrete AMD workstation is the lack of proper rr[0] support[1] in their CPUs, and for the GPU the rather poor hardware encoding[2] that makes it infeasible to seriously stream out 2160p60 without problematic visual artifacts.
[1]: https://github.com/mozilla/rr/issues/2034#issuecomment-58361...
[2]: I have not found serious competitive benchmarks for the recent AMD hardware encoders. I would appreciate such.
The reason I'm asking is that's not what I expect from my experience. The main issue here is the CPU necessary to get equivalent quality x264 encoding? I would expect x264 medium profile to be better quality than any (consumer) hardware solution and I can do x264 1080@30 medium encoding in real time using ~70% of my old 5820k CPU. A modern Ryzen (3600 or higher) would use even less CPU so really not an issue, for a single stream (but it seems Nvidia has artificial limits to the number of concurrent streams anyway).
If it's a desktop APU then it has the hardware.
Keeping it hot basically won't matter unless it already had defects.