AAV under the sink - the real cause of drain smells or a red herring?

I had almost this exact thing in our powder room and it turned out to be two seperate problems stacked on each other. The smell right after running water was gunk sitting in the overflow channel of the sink. Every time water hit it, it stirred up the crud and pushed that sour smell out. I pulled the pop-up stopper, fed a skinny bottle brush up through the overflow hole with some diluted bleach, flushed it a few times, and the odor dropped off dramatically within a day or two. If you've never cleaned that passage since the sink was installed, I'd bet on it before blaming the plumbing layout. The gurgle is its own thing though. Ours turned out to be a sticky air admittance valve that wasn't opening fast enough when the sink drained. The plumber who swapped the old one out probaly fixed part of it, but cheap replacement valves can be junk out of the box, and one mounted under a cabinet in a closed space seems to gum up faster than mine did in an open utility area. Easy test before anyone rips anything out: fill the sink, pull the stopper, and watch the trap with a flashlight while it drains. If the water level visibly dips way down when it gurgles, you're pulling the seal and the vent side is the culprit. If the water stays put and it still gurgles, I'd stop chasing the vent and go after the drain guts instead.

I'd pump the brakes on ripping anything out. I ran an under-cabinet air valve on a basement bar sink for over ten years with zero smell issues, so the setup itself can work fine when it's done right. Mine only acted up once, and it turned out the valve I'd bought at the big box store was a cheap knockoff that stuck partially closed. Swapped it for a Studor brand one and never thought about it again. One thing worth checking before anything drastic is the height of that valve. Mine had to sit a good bit above the trap arm to work the way it's supposed to, and I've seen them mounted low under sinks where they can't do their job. If the plumber swapped the valve but never looked at where it sits or how the pipe runs to it, the new part didn't fix much.

Also, the gurgle after the water drains is your trap getting pulled on, that part of your theory sounds right to me. But a slow draining fixture with a clog partway down can make the exact same noise, and the smell could be sewer gas or just rotting gunk in the line. I'd snake that drain first, it's a fifteen minute job with a cheap hand crank auger, way easier than opening up a wall. If the valve turns out to be mounted wrong or undersized, fixing that is a Saturday job, not a reroute.

Honestly, those small under-cabinet valves are always a compromise. Back when we installed things, we ran proper venting all the way to the roof, full stop. Those little local vents aren't built for the heavy backpressure of a whole plumbing stack, expecially if the main sewer line is getting overloaded somewhere else. I've seen this happen before. If you've cleared the trap and the drain guts, take a look further down where that line connects to the main stack. That's usually where the whole system starts struggling with air pressure.

I agree with the point that the under-cabinet valves themselves are often the real problem. It's crazy how many of those cheap replacements are made with poor quality plastic, regardless of the installation setup. I had a setup like this years ago in a laundry room, and even though the drainage seemed fine, the air admittance valve started weeping a little bit of water every time the drain activated, which was a clear sign it was failing to seal or operate correctly. You really have to check the valve function under the cabinet before assuming the entire pipe system is broken.

First post here so hopefully I do this right. Our situation was a little diffrent but maybe relevant - the smell only showed up in a bathroom nobody uses much, and it turned out the trap water had just evaporated away. Flushing it regularly fixed it. So if this guest bath sits idle a lot, it might be partly a dry trap thing on top of whatever else is going on.

Also that flashlight test the earlier poster suggested is a good idea, I did something similar with ours and it saved me from replacing parts for no reason. Does the cabinet door stay closed all the time? Wondering if that boxed-in spot makes the valve struggle more.

One thing worth checking is wether the valve's actually sealing shut when nothing's draining. Those things are one-way, air in only, so if the diaphragm gets crusty it can leak sewer gas right there under the sink. That'd explain why the smell lingers even after the swap. Ours did this in the hall bath and I only figured it out by sticking my nose near the valve itself. Also make sure it's sitting vertical like the instructions say, they can stick if they're tilted.