The argument has major problems with it, but there’s at least support for the first piece, i.e. that homosexuals do tend to have more mental health issues.
Not being an expert on the data, I have a simple issue with this: The issues being complained about (possibly legitimately, depending on the data) are very likely to be caused by the discrimination itself.
You can’t call someone a freak for their entire lives, tell them they’re unnatural, tell them God hates them, and then count it against them when they end up having issues. That’s ridiculous.
Equality first. If, after some number of decades of normalization, we find that there are still significant issues with mental health between the groups, then let’s tackle that. But until then, don’t be the idiot that advocates institutionally causing the damage that we plan on blaming them for later.