This isn't a direct claim of fact, but...

It seems very unlikely to me that a UCMA enthusiast would grant that a UCMA has in any given case only a fifty percent chance of being UC. So to assume this begs the question against them. It may be that the UCMAist is being silly here, or that the burden is on them to show that things are otherwise, but that's not relevant to the question of the strength of EY's argument against UCMAs.

...I think this hidden assumption is a very reasonable one.

It is, but it's a bit too reasonable: that is, it's unreasonable to think that the UCMAist actually thinks that the UCMA is already explicitly accepted by everyone, or even that everyone could be immediately or in any circumstances persuaded that the UCMA is true. UCMA's on this conception are obviously false, but then EY's argument is wholly trivial. Nor would we need an argument: it is not hard to come up with a single case of moral disagreement, and that's all that would be necessary. But this would be to attack a stawman.

The UCMAist is committed to some sense being given to the UC bit, you're right. If we go to an actual UCMAist, like Kant, the explanation looks something like this: People say all sorts of things about their moral beliefs, but no one could have reasons to doubt the UCMA while holding consistant beliefs. This means that in principle, any mind could be persuaded to accept the UCMA, but not any mind under any circumstances. I (Kant) am committed to saying that every mind is so structured that the UCMA is an unavoidable deductive conclusion, not that every mind in every circumstance has or would arrive at the UCMA. So if this is what being a UCMA means:

"A(m) is true if mind 'm', when presented with the argument A, will change some specific belief / internal state / thought / behavior to state Y"

Then yes, UCMA's are impossible. But no one has ever thought otherwise, and it remains open whether something very much like them, namely moral arguments which every possible mind is committed to accepting (whether or not they do accept it) is possible.