for fun, i just wrote an autohotkey script that does this. but it's probably not what i should use because it writes to a file every minute. and i think that might damage my hard disk drive.

at the push of a hotkey, it reads the title of the desktop icon that is selected, pulls up the time logged for that game from a file, then writes the time to a text file which obs displays on stream. the time updates every minute. i'm worried about constantly using the disk drive that way, and i definitely don't think i should make it write every second instead.

does this already exist, in livesplit plugin or something?

There is this: Not sure if it's exactly what you want, as this only shows the time spend in actual runs on that exact splits file.