Yes, the claim that ‘not allowing gays to adopt would leave children homeless’ absolutely IS false.
It’s been covered upthread, and you yourself acknowledge, that there are numerous problems with the SYSTEM itself. This is not about a lack of heterosexual married couples wanting to adopt.
The
US Census found that less than 1% of households are gay couples. There are many reasons I would guess that a lower ratio of gay couples are seeking to adopt children. So, I"m sorry, but no, there is no way that 1% or fewer are the ONLY thing standing between children and homelessness.
We might consider, too, that those who claim to care so much about children were willing to shut down Catholic adoption agencies–in other words, give children to two gay men, or we’ll shut you down and hurt ALL the children you could have been helping. Never mind that other agencies were willing to place children with gay couples. Obviously, there’s more concern out there about the gay agenda than about the children themselves.