Cannot start MySQL - server quit "w/o updating PID file" (PID file missing)
I solved it! When I finally got it to write an error log (needed to fix the permissions for that) and looked inside the error log, I saw it was missing the symbol _clock_gettime(). When I googled that, I discovered MacOS didn't add that symbol until OSX 10.12, and I'm running 10.11 (so as not to break Dreamweaver CS6).

I uninstalled MySQL 5.7.21 and looked for the latest version of MySQL that supported El Capitan (10.11), which turned out to be 5.7.16. I installed that, and now it's running fine.