Why is there no 'Rename' option when right-clicking an item in Final Cut Pro's browser?

You just press Enter, just like in Finder. It is not necessary to right-click and select rename. This is documented in the Mac User's Guide: Rename files, folders, and disks on Mac - Apple Support

The "Enter to rename" behavior began with the Lisa Office System in 1983 and was adopted unchanged by the first Macintosh Finder in 1984. It has remained in every subsequent generation of the Mac operating system. So it's been this way for over 40 years.

Applications that embed a Finder-style browser (Final Cut Pro, Logic, Numbers, etc.) inherit the same behavior automatically. Any third-party MacOS apps that also use the framework calls NSOutlineView / NSBrowser will likewise inherit the system’s “Return → beginEditing:” action.

I think the original UI design reason was that Finder was a file manager, not a file launcher.

You can simply click the Name in the browser and then tab to the field you wish to rename, no waiting. This can be done in rapid succession just using the keyboard Tab, Return and Arrow keys. Or use the Inspector window to rename a single field or in batches many of the common fields such as Name, Notes, Reel, Scene, Take, Camera Angle and Camera Name. Just make sure the Inspector is set to Extended to see the most options.

I simply click to highlight and then click again. Items in the projects column only need one click.

It's pretty much instantaneous.

Incidentally it works the same in Resolve so it appears there is no demand for it.