Switch streaming audio issues

Such bullcrap.

The goal is to stream and play Switch games while listening to audio live. The simple solution would be to plug Switch HDMI into my capture card, HDMI passthrough from the capture card into my monitor, and headphones into my monitor. This way I capture audio to the stream and I get to view and listen to the game with no delay.

That doesn't work for two reasons. The Elgato Game Capture HD is notorious for randomly desyncing audio, requiring constant resetting of the device in software. Additionally, when HDMI is plugged into my monitor from the Elgato, no audio is passed through. I can plug HDMI directly into the monitor (skipping the capture) and get audio, or I can plug the Elgato output into a laggier TV and get audio. However, for whatever reason, my monitor and the Elgato do not talk audio to each other.

So I tried instead to use the Switch's built-in headphone jack to get audio. I can plug my headphones straight in and get audio for myself, but the stream gets no audio that way. Should be able to use a splitter right? Wrong: the way the Switch headphone jack is built, most of my 1/8" aux and audio cables do not fit properly into that port.

But surely some of them do, so I should just get an extension, split off to RCA or 1/4", and then run that to my mixer. In theory, that lets me balance game audio with my microphone and easily manage voice in my computer. I can use the two different outputs of my mixer so that the stream gets game and mic audio, but Discord gets only mic audio. Indeed, this is what I do when I capture using composite from a Wii, again to avoid audio desync issues in the Elgato and to give me these added bonuses.

Except the Switch has a headphone jack, not a line out. This creates a ground loop when I adapt it to plug into my mixer, which causes a super loud and annoying buzz. In order to get clean audio from my Switch to my mixer, I would have to add either a ground loop isolator ($10, not even sure it would work) to the Switch headphone jack spaghetti, or an HDMI audio extractor ($25) to the Elgato spaghetti. Not sure which I prefer.

I could play on the laggy TV and plug my headphones into that, since for whatever reason the Elgato will then output audio properly. But again this introduces chronic desyncs to my stream capture. I have no clue why my main monitor does not accept audio from the Elgato like my TV does, despite the fact that it accepts audio from any other source over both HDMI and DisplayPort. There exist no forum posts or support pages which describe the issues I'm having.

I blame Elgato.