Accidentally removed /usr/bin - is my Mac permanently damaged?
You cannot permanently damage your computer just by deleting files. At worst, you will render the OS non-functional.

Just run the macos installer of your choice that is the same version (or newer) as your current os. This will fix everything needed to boot your Mac and not overwrite any user settings or programs or files. 

If you have deleted only /usr/bin, (NB: not /user, which does not exist) then you should be able to restore this from your backup. It's normally hidden in the Finder, so press Command Shift dot to show hidden files in the Time Machine window. 

If you think you may have deleted other files and folders, you would be well advised to re-install the OS. You should be able to download it from the Mac App Store (depending on what version you're on), and then just launch the app. Your user files and preferences will be maintained.

Deleting system files is never a good idea: normally, you would have had to have bypassed a number of safety features and warnings to do this. If you want to ask another question about what you were trying to do, you may get an answer that provides a safer method.