Step one, wire brush to get the surface crud off. Step two, PB Blaster--and the can fell off the dryer and the friggin' nozzle broke off, and worse it's one of those nozzles you can't replace, so I now have a nearly full can of PB Blaster which is almost totally useless. I just stuck the stub on the fitting and pressed, spraying PB Blaster everywhere.
Still wouldn't budge.
Step three, Dremel with cutting wheel to make a notch in the hose fitting. Step four, screwdriver in the notch and tap with a hammer (I used the wire brush handle). That split the thing enough to let me unscrew it. I'm pleased to say that I didn't even nick the threads on the spigot with the cutoff wheel; I was careful.
Step four: water weenie into pipe, garden hose from spigot to water weenie. I used the 50' nylon garden hose, the one that reels up. (Note to self: see about making a shorter hose, maybe 10'.)
Step five, turn on water. Turn off water, go upstairs to check sink, find that sink is now full of somewhat dirty hot water; plug sink and try again.
Step six: success.
The water weenie did enough; I let the water out of the sink and it drained in reasonable time. It is still "leveling", where the water from the full side will backfill into the empty side, but that's because the drain is not completely clear. It is draining, at least, and I was able to wash a week's worth of dishes without any trouble whatsoever.
I will have to go back and redo the snaking, and I'll have to redo it slowly and carefully--and maybe make some kind of shredder/cutter thing to screw onto the end of the snake to bust up the grease--but the water is flowing again, and we can use the sink, and that's the important part.
I let full hot water run into the sink after the dishes were done, and added more dish detergent. Maybe that will help.
Anyway--not too bad, about $30 invested, and it's working well enough that we can use it until I have time to really clear the pipe.
A plumber would probably have charged $300. He would have got it completely clear in a couple of hours, but then a plumber has years of experience and all the tools that I don't have. For $270, though, I think I can take this outcome.