I tried all of the suggested options, and a few more found at this link. No luck.
Then I tried to add a reference from a Web Site (as opposed to Web App). The process is different: you have to right click the project and go to Property Pages, and there's an Add... button on the References tab.
It still crashed, but there was a message in the Event Viewer this time that pointed me to the full path of a DLL in a 3rd party component I have installed. The DLL couldn't be read from disk (corrupt). So, a repair of that lib and a reboot later, and I'm back to good.
UPDATE:
I came to find out the real reason, several files on my SSD had become corrupt. CHKDSK /R got me back going for a while. Eventually had to replace the drive. Just a reminder that it may not be VS's fault.