The desktop Plex player failing to launch for me turned out to be a file called ‘highlander.lock’ in “Users[user]\AppData\Local\Plex”. Once removed, latest version of Plex (currently v1.39.1.2763) launched normally. Deleting the entirety of that folder folder was not necessary. The file in question was not recreated, no idea what it’s purpose was. But this fix worked on multiple machines I own, running both Win 10 and 11. Prior to removing the file only the older version of 1.29.x.xxxx would work.
(AppData is a hidden folder within your user profile directory. As others have mentioned, typing %LOCALAPPDATA% in the Start menu will give you quick access if you don’t want to deal with unhiding these folders)