you don't necessarily need complexity of HDMI to audio extractor

I didn't think it was all that complex. HDMI passes through unchanged, including audio, and another copy of that audio squirts out the side. Done!

Doesn't do anything for a ground loop: that possibility remains the same either way. And just because you have a digital signal, does not mean you're immune to them. As soon as you convert to analog, whatever ground loop noise you might have gets in *there*.
(This includes a USB connection to an otherwise analog mixer, which is simply a standard USB soundcard inside the case. If the USB connection completes a ground loop, then it'll buzz like anything else. Do that with an all-digital console, and it may not because it *stays* digital.)

older, especially analog, system perfectly adequate for Sanctuary use, but doesn't handle things like Compression

I know what you mean, but... ;-)

Mostly inherited from various church upgrades, plus a custom cart. (woodshop out of frame to the left)
40-channel console, 16 channels of outboard compression/gating in the left rack (2x ACP88), video soundtrack on the two matrices and then through a 2-channel compressor at the top of the right rack (Alesis 3630), and then to a Behringer USB line-in. Below that is a crossover for subwoofers, post-everything. Center rack has a 2-channel FX unit and 24-track recorder/playback. Between the rack and the console are 40x custom highpasses to replace the console's 100Hz 18dB/oct yes/no with no other options, and two of them towards the right (FX returns on full channel strips) are using their option for a lowpass too.

Pure analog, except for the FX unit, the recorder that does no processing whatsoever, and the soundcard to OBS that also does no processing whatsoever.

The picture is when I was still building it. It's now temporarily replacing a Midas M32 while we change some things around, and they love it! (some have asked if the M32 is going to sound like that after we switch back...) Video sounds decent too...after a couple of services to listen back and tweak the matrix sends for next service.

Anyway, back on-topic: :-)

if sound system doesn't have available input

Then I'd say you need a bigger board. :-) Nothing wrong with having unused inputs, but swapping plugs mid-service, inaccessible submixers, and bypassing the board altogether, all bring their own problems and ballooned complexity, at least in part because they create so many exceptions to something that could have been a universal rule.

If space is a problem (shortsighted design, or never intended for the present use in the first place), there are compact digital mixers that pack a LOT of channels and processing into a small space...with the tradeoff of only seeing a tiny little part of it at any given time. Unlike analog, which puts everything in front of you all the time, a paged digital board requires you to already know what else it does and what page it's on.

PowerPoint and OBS Studio on same PC

I can see that, for screen capture and looped-back audio inside the same box, for perfect reproduction with no external wiring. But I still see a big operational benefit to having a dedicated machine for each task, even if the same operator runs both/all of them. Conceptual boundaries like borders on a map, simultaneous cues without mouse contention, fault tolerance, etc.

And like the HDMI audio extractor, I didn't think a physical wire from GPU out to Capture in was all that complex. Even if there's a multi-in/multi-out matrix switch/splitter in between.

Conceptually, "this machine feeds that display." "Okay, got it." "Now tap off of that signal, and also feed it to this machine to put in the corner of the frame, and here's the wiring that does exactly that, verbatim..."

If there is a 'sound engineer' manning the sound system, great, but that is often NOT the case. In that case, having separate inputs into OBS Studio might be necessary so OBS Studio operator can control the various inputs

I'd call that a strong case for a remote-controllable digital mixer of some kind. FoH Engineer or not, the Broadcast Engineer can still remote control that separate mix in the console, and so everything stays routed through there.

I once had the idea to mix the video sound on the main bus, and FoH on some post-fade aux'es, with the reasoning of:

• It's actually FoH that needs the "weird" mix because of stage spill and acoustics, while the video is actually the "sane" one.
• The video is pretty much always stereo full-range at that point (no center or sub), which matches most main busses, while FoH is, again, a weird combination of things that can easily be accommodated by a series of aux'es as needed.

The crew rejected that idea almost immediately.

I guess the common thread here is, "*operational* simplicity, at the eager expense of some additional boxes and wiring if necessary." All the controls for each concept are in one place, and each concept has its own separate controls. But hard-reject something that promises to do EVERYTHING all on its own, because it very likely won't (designed for someone *else's* "everything", not yours), or will fail at some point as an unfixable black box. Make it obvious, or easily intuitive once shown, how it has to be, so that troubleshooting becomes easy too...for someone who doesn't demand a brittle list of specific mouse movements and reject any semblance of actual understanding.