You’re quite right that it’d be very hard to demonstrate a causal effect (and without having read the study itself—don’t have access -, I suspect the researchers didn’t even want to try).
Actually, I have no idea how that could be done in practice. For voluntary hospitalization, it would be helpful if one couldn’t be hospitalized against one’s will, but I’m not aware of a time and place where that would be the case, and don’t expect there to be any. So one can study that only outside the realm of “hospitalization-worthy” suicidality, by using patients who have been offered hospitalization, but declined it, as a control. My quick search turned up no indication of even that having been done.
And for forced hospitalization, it seems sort of impossible in principle to find a control group...
I also just found this, again I don’t have access, and unfortunately it doesn’t even have an abstract. Might be relevant, though, judging by the title.
There is one study I’m aware of.