My setup looks like this:
OSSC -> AVR -> HDMI Splitter with Audio Extractor -> PC with Optical Input (via separate Soundcard)
I’m also getting a little noise when the game is completely quiet, while there is no (noticable) noise over my home theatre, but I’m certain this is because of the analog to digital conversion which is inevitable.
Your best bet is equipping your consoles with an optical audio output, which is pretty simple to do on most consoles. Then you’d have to get a Toslink switch, theoretically there should be absolutely no noise with that setup.
For merging the audio with your PC you’d just have to get a cheap soundcard with an optical input, I’ve had success with a 20 buck soundcard from LogiLink (this one). But since this soundcard has problems with some encodings specifically the encodings of most games on the PS3, I had to get an older internal PCIe card (Creative X-Fi Titanium) which works with every console I have. And since the SPDIF input is listed as recording device, you can simply mix it with OBS so you can easily use it for streaming.
The newer Creative Soundblaster AE-5 and AE-9 also have optical inputs, while they aren’t cheap, they provide pretty good sound quality and have replacable op amps, which might be worth considering if you don’t have a good DAC+AMP combo already.