You lot seem quite knowledgeable. I have a Grohe shower, which is a horizontal chrome bar about a foot wide. It uses a mechanical cartridge to mix hot and cold water. Over 20 years it now is running very cold and I am sure there is a good supply of hot water.
Looks like a cartridge replacement is called for. Looking on Youtube, this does not seem a difficult job (*)
How to identify the correct cartridge? I guess take it out and look for a part number?
(*) Hah. This from the idiot who unscrewed the pressure relief valve to his apartment. And failed to get it back on.