Funnily enough, to me "what else" sounds more like "what would you like to do in addition" as opposed to "what would you like to do otherwise".
Certainly in Australia, "anything else with that" is quite common in shops, particularly in food shops, or pubs. Or at least it used to be be when I was there.
Sure, but if you ask for A, B and C and someone asks "what else?" then they mean "what about things that aren't A, B or C?" It's still the same basic meaning, set complement, but about things rather than logical states.
The question isn't so much whether "else" is used at all, but rather whether it's used as a conjunction. That's why the article author mentioned "or else": here "or" is the conjunction, so that proves "else" isn't fulfilling that role. In this example you gave:
> What else do you want to do?
"else" is acting like a noun rather than a conjunction (compare with "what activity do you want to do?).
Certainly in Australia, "anything else with that" is quite common in shops, particularly in food shops, or pubs. Or at least it used to be be when I was there.