I've lined out the broken tub spigot replacement on the To Do list! This was a real tale of triumph over adversity and uncertainty.
After some delay tactics doing chores that I know are well within my competency - replacing the kitchen sink washers to halt a slow drip there - I started with vise grips on the rusted nipple. They just crushed it down and I got nowhere. The pipe, normally a thick sidewall that could take this, was lace thin. I then took a hacksaw to the crushed pipe end and flared out the remaining stub with a large persuader - enough to jam a nipple extractor inside. I tried this tool on the advice of my local hardware store, but had no luck - the pipe was too corroded inside for the nipple extractor to grip well.
Finally I returned to the hardware store where I ended up with a #7 Easy-Out which did the trick after applying a whole lot of muscle.
Rex from Rare Earth News and Ballona Native Plants Compendium, who left me advice below deserves a huge thank you for his suggestion to use an easy out. The corrosion was localized to the spigot pipe nipple only, so I was able to just replace it with confidence that the rest of the pipes won't break similarly. I believe that the failure was brought about by corrosive cleaners, which got in under the spout and sat on the galvanized pipe, rapidly turning it to crumbles. This was aided by a leaky faucet (which I still need to tackle). Since this is my only bathroom, I felt extremely fortunate that I didn't disable it for more than a few hours. Juli advised me not to take on the faucet, and I have to agree that was the right choice for now; they are so corroded that I can't even get the handles off!
The new spigot doesn't mount flush to the wall because the "homeowner special" tile installation is waaay off vertical. Not to mention the spigot pipe doesn't come out exactly straight Neither of these are correctable right now, nor are they my fault. The bathroom needs a redo in the next couple years, so I'll get it then. For now, a healthy glop of caulk will hide it.
Photography credits: Juli.