Imagine trying to play online with this; fighting input lag along with actual lag. It'd be enough to drive a person insane.

The reason why CRT's are important for Melee is because of input lag.
Input lag is already inherent to HDMI, you are getting worse lag if you are using that AND a converter.

Maybe that's the joke, but as someone who lurks around the CRT community and see people trying to play modern games on CRTs because they only want to game on CRTs… I doubt it.

Solution: Get an HD CRT.