Ah, but you can't. You need to play a card while playing a spell, which you can't do - and nothing's giving you the ability to.

Just like nothing gives you the ability to spend while you play a spell?

I fear the ability to cast other spells might, terrifyingly, work. But I'm pretty sure you can't play lands while there's anything on the stack, and this card is on the stack by the time you're paying its costs.

(EDIT: I was talking rubbish. You certainly can play a land if something lets you, as long as it's your turn and you haven't yet played a land.)

If you can't play lands while something in on the stack, Brilliant Ultimatum didn't get the memo (that's the whole reason it uses "play" instead of "cast").

Brilliant Ultimatum works because it explicitly tells you to go ahead and do it. I'm not sure this does - it just says "you can't play me unless you do".

Would it actually lose anything to have instead "You can't cast ~ unless you've played another card from your hand this turn"?

The fact that you're told to pay it as a cost is what gives you the power to do so. Nothing in the rules lets you remove mana from your mana pool at an arbitrary time, or reduce your own life total voluntarily, but costs involving those things obviously work. It's the same concept here.

The reason I don't use that other wording is because it allows the island you just played on turn 1 to qualify. Change it to nonland card and you have the forgettable Skyshroud Condor.

Umm, shouldn't it have flying? Just sayin'.