Can a washing machine recover after sitting shut for months?

Savable, almost certainly. The smell that keeps coming back after a day or two usally isn't the seal itself, it's the drain pump filter behind that little hatch at the bottom front. That's where all the grey sludge collects, and since the machine sat with water in it, that filter is probably disgusting. Open it slow: put down a couple of old towels and a shallow tray or baking dish under it first, because a surprising amount of water comes out even on a machine that seems empty. Unscrew it, pull out whatever's in there, rinse the filter under hot water and scrub it with an old toothbrush. Wipe out the recess with rags while it's open. That alone usually kills the lingering smell. While you're at it, pull the detergent drawer completly out. Most of them have a little release tab near the softener section. Wash it in the sink and wipe the cavity it slides into, because that area grows its own science experiment too.

For the seal, the trick is to peel back the folds of the rubber gasket with your fingers and wipe inside the crevices, not just the face of it. The slimy film does come off, but it usually takes a few sessions over a week rather than one wipe-down. A rag with diluted vinegar works, just don't soak it. If after a couple weeks there are still black spots actually embedded in the rubber, those are mildew stains that have set, and they'll smell faintly whenever the drum's damp, but honestly if the laundry comes out clean and the machine smells fine once the filter's done, I wouldn't replace the seal over a few cosmetic spots. After that, run another hot empty cycle with your cleaner, and going forward leave the door cracked open between washes. That's the whole secret to keeping these fresh.

That filter hatch is where I'd put my money too, been there more than once. One thing the younger crowd forgets is the old trick my mother used: a scoop of washing soda in the hottest cycle instead of fancy cleaner. Works just as good and costs pennies. Also check where the drain hose hooks into the wall pipe, that connection can hold a sour smell and nobody ever looks back there. And don't forget to prop that door open every single wash day. Machines sat shut for months always stink for a while, mine took a good month before it fully aired out, so don't panic if it lingers a bit.

Gotta push back on the 'live with a few cosmetic spots' take. I tried that exact approach on my old machine, figured stained rubber was fine as long as laundry came out clean. Two years of cracking the door and running cleaner tabs and the smell never fully went, it just hid until humid days. When I finally priced a replacement gasket it was way cheaper than I expected and swapping it took maybe half an hour following a video. So if those black spots are actually set into the rubber, I'd say just order the new seal instead of chasing the smell around for months like I did.

On the seal film, dilute bleach on a rag was what finally got mine clean after vinegar kept letting it creep back. Machine came out of a rental years ago with the same slimy rubber and it took two or three goes, wiping inside the folds and then going over it with a damp cloth after. No smell since, and that was maybe four years back.

One more spot nobody's pointed at yet: feel under the bottom edge of the door boot where it meets the drum. Stuff hides down in that lowest fold and the water never fully drains out of it, so it stews there between washes.

It's savable, been through this exact rescue a few times now, seven or eight machines pulled out of basements and rentals over the years. One spot the filter and seal advice misses is under the drum paddles. Those plastic lifters just click in and the slots under them collect hair and sludge that never gets rinsed. Give each one a firm yank toward the door, most pop off without tools, and clean the gap underneath. If the smell survives the filter clean, that's usually where it's living.

Also worth a proper maintanence wash once a month after this, hottest cycle your machine has, empty, maybe half a cup of washing soda. Cheap insurance once you've got it smelling right.

Update: finally opened that filter hatch and wow, the second poster called it. Grey sludge everywhere, pulled the filter out, rinsed it under the tap, ran a hot wash and the smell is mostly gone. Also tried the dilute bleach on the seal folds like suggested and the film came off after two passes. One thing, the reply about checking where the drain hose hooks into the pipe cut off mid sentence. What was the rest of that?