Are there downsides to quitting the Finder after enabling the menu item to quit it?
Finder is the application Mac uses to well... find files.

It should be perfectly fine so long as you do not run anything that specifically calls finder, i.e. AppleScript. You'll also (obviously) lose all your open Finder windows. 

I completely understand quitting for presentations, but there should not be much of a performance improvement if you quit Finder.

So, no, it should not be a problem, just note that apps that call on Finder should open Finder for you, so you'll need to quit it again. 

Thanks to @user3439894 for catching me on something I said wrong.