One of the great things about this forum , the current original poster will find, is that there are knowledgeable discussions about things that one may not even have thought was a question.
The lack of multi channel inputs is a problem here at QQ. Unfortunately it is NOT a problem for the manufacturer's of retail consumer equipment because they don't care about Quad vinyl of any stripe, nor of people using Surround masters, nor even do they seem to care about people playing DVD-A and SACD and BD-A. (and preferring to use Oppos excellent D/A converters rather than the ones in receivers and processors).
You are not going to be able to go from four channel RCA (a format that was commercially obsolete in 1976 or so) into HDMI. If you want to play your quad LPs you will need to get something that has analog inputs. Like say a ten or fifteen year old Pre Processor. You could then get a two matching stereo amps and put those speakers on speaker selector switch boxes.
I note that you are in St. Louis and I have a SchMarantz AV7005 pre processor that is at the moment , not hooked up. I might be able to be lured into bringing it over.
Also if there was any chance of doing what you are trying to do, it is likely that someone here would have already tried it. However from time to time we have posts and inquiries and things we would like to do that just are not doable. I think your situation falls into that unfortunate category.
I agree it's likely a lost cause with HDMI solutions costing hundreds of dollars... but I think I figured out a 30 dollar solution that works just as well to use the same speakers without having to unplug in and out everything. I found a switch box that strangely enough is a 4 channel one! Why would anyone need one of those lol? I'll have to run it backwards technically but this should be more of a direct wiring solution vs trying to hotwire an HDMI cable. I just hope it fits the 12 gauge wiring!
It ain't over till I get a little electrocuted!
Long story... I need to add to my profile all the hardware I've picked up in the last month to try to get play out some quad.
1. Started with a Panasonic SL-701 Automatic 4 Channel Quadraphonic Turntable CD-4 with built in demodulator.
2. Then a Panasonic RS-888S 4-CH All In One Stereo System which I mistakenly thought had a demodulator in it for the turntable, but it only has one built in for the 8 track player. My bad! I may not actually need this anymore so that's 200 bucks gone to the wind.
3. Lastly - last night - stumbled on a Lafayette LA-975 SQ demodulator with only a stereo amp but quad outputs. That led me to research that brand and from this forum found a 1973 474 page Lafayette catalog, see it here
https://device.report/m/425e81ba46c1e0fd168ffffd06a6a6f1bcd16f3aec343bc51d3b165e5335fcb3
So I searched some of the SQ amps in that and found on page 9 the Lafayette LA-64 amplifier which is a full quad amp with SQ-L, they had cheaper units back then with SQ-M, and a few years later they came out with SQ-W which were regarded as competitors to Tate's, which people then say Involve Audio sweeps those out, but I couldn't find any of those for sale under 500+ and at that price I was thinking I would be better to sell off my Onkyo and buy a Marantz with the 7.1 CH inputs. So that's 3 new big devices, I'm just waiting for the amp to come in.