This a topic I am very sensible to. In Brazil, at least, this is not the case. I travel around the country and this is something I always pay attention to. In my personal experience, they are not over-represented.
I was lazy in my wording. I meant that transgender people are over-represented amongst the homeless population in NYC compared to their percentage of the population. Sorry for the confusion.
Obviously, I can't speak to what is the reality on the ground in Brazil.
Anyway, this is secondary. "They deserve shelter" is trickier than it sounds. I would phrase it as "I wish no one was homeless and I would love to help".
Homelessness is a very complicated problem. One part of it is that a lot of homeless people chose to be homeless, consciously refusing shelter. There are the severe drug addicts, there are the psychiatric patients. It is an extremely complicated issue.
I agree that homelessness is a complicated problem. Which that little video does not address. Yes, some people prefer to live on the streets. If you ask them why, many will say that they are concerned for their safety in a shelter, which is why segregated shelters exist. Many are drug addicted, and many have mental health issues. Often, these are co-occurring issues. Sometimes mental heath problems lead to self-medicating with drugs, and, conversely, drug use can create mental illness. You can't force people to take their anti-psychotic medications, and you can't commit them forever against their will, so, often, they end up (back) on the streets. It's VERY complicated.
Having said all that, the initiative above sounds like waste of money and energy. But I won't lie, I haven't watched the video. Is it very likely that the person who created has trans prejudice? Yes. Is it likely he _still_ has a point? Also yes... That's the crazy world we live in.
I actually wanted to say this: I definitely think that if the shelter were not created for trans- and gender nonconforming people, there would be no video. And certainly Kieran wouldn't have posted it. Who cares how much NYC invests in a homeless shelter?
But why would it be a waste of money and energy? It will certainly get some people off the streets. Some of that "extravagant" money is for mental health counseling, drug counseling, and efforts to get people mainstreamed into jobs and society. To help find permanent housing. Homelessness is not just a problem for the homeless. It's a quality of life issue for everyone who lives with it. The fewer people on the streets and sleeping in the subways, the safer we all are.
I absolutely know we can't save them all. But it's worth trying to save the ones we can. You'll entice more into care by creating a nice facility, rather than the shoddy, unsafe, communal facilities of old. Just because you're poor, doesn't mean you don't deserve to be treated with dignity, in one of the richest cities in the world.
I do some work with a tiny little church in my neighborhood. They have a soup kitchen they can only afford to staff and keep open 3 days a week. What do people do the other 4? They find housing for people, usually men, one person at a time, and take donations for the things they need to start a home and a life. It's a very one-to-one effort. All (or almost all?) volunteer effort, with the kindest people I've ever met. But it's a drop in the bucket. Cities, states and the nation, as a whole, need to do the heavy-lifting.
(Also, that's Blowhard in the traditional sense, just in case 
