she's already done one strategy by placing actual text in the button describing its function. the other commented thing here is aria-describedby but that's intended to be a more verbose version of aria-labeledby. neither of which tell of any action like aria-controls does
having low support doesn't make it wrong, it means those supposed to do the support are lacking. there's always a choice of how much and how far to go. between the added text and the aria-controls anything further is overkill and possibly enabling future lack of support by creating less demand
but it's a button and this is a descriptor, at some point we're literally arguing semantics... which is good but I've likely already gone farther than i should
I am commenting on it because this is a public context, and that folks less familiar with it may extrapolate on its functionality based on your comments, and then use it without testing.