(homebrew.mxcl.mysql[2274]): Service could not initialize: Unable to set current working directory. error = 2: No such file or directory

You probably enabled the MySQL launch agent with brew services start mysql in the past and forgot to disable the brew service (before uninstalling mysql) which would remove the plist.

So unload and remove the agent from the launchd database manually with:

launchctl unload ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist
launchctl remove ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist

Then remove the file:

rm ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist

If you did a custom install of the plist, the file homebrew.mxcl.mysql.plist may reside in /Library/LaunchAgents. Use the same commands then but adjust the paths above and prepend sudo.