Is this worded correctly? Does it even work?

It.. probably does work. Might want to be 'you have a' not 'there is' but I'm not sure.

It's kinda demented; I can half see "Why would I ever want that?" and half "Yay! All my end of turn mana I saved I can use now!"

So it's actually, kind of, "You may untap at the end of your turn, instead of the start, if you like"

Madness!

I think the wording works. The closest analogy is "After this phase, there is an additional combat phase." Possibly it should be "before each of your untap steps", I'm not sure.

IIRC it's roughly equivalent to Leyline of Anticipation, it lets you cast sorcery-speed spells at the end of the opponent's turn, after you see if you needed to cast any counterspells or kill any attacking creatures. (But doesn't also let you flash spells during combat.)

Which is definitely something people would like to do, but I'm not sure if it's worth moving the phases around to get it or not. There may be some other cases where it matters, but I can't think of any.

Casting spells before 'at the beginning of upkeep' sounds useful, if not rare. I wouldn't mind the chance to cast an gating creature to return a demon before he deals damage to me. Or cast Monstrify on a creature before Phthisis goes off...

Yeah, this is subtle but does actually interact with a surprising range of cards.