Are non-religious homeless shelters any better/worse? Obiously not all christians are saints and not everyone who calls himself a christian follows christ.
Most homeless shelters that I'm familiar with are in the US, and most of those are Christian. I really couldn't tell you. I stopped hearing so much about them when I moved to Northern Europe. I don't know if it is a language barrier, the culture, or if they are better.
I'm not going to distinguish between the folks that "follow christ" and who does not - many are going to claim it and I'm not really willing to engage some of these folks to see if they are "true" or not. Living in the US, there wasn't a big outcry against folks discriminating. You were told to just be normal. Don't be gay. (I'm bisexual). In the 90s and before, a lot of folks used it to justify outright racism. And there isn't enough outcry now or then.
Granted, this isn't just Christianity. But I'm not going to give Christianity a break because they aren't special when folks are glossing over the harms.
> Are non-religious homeless shelters any better/worse?
Well, I have worked with organisations here that originated from the communist party (in a Western country without a "complicated" history with communism) and they just don't discriminate at all. Not even against people who might be "not poor enough" to actually need them. It seems to me quite better indeed than Christians denying help to trans or gay persons.
Ah, well each new brand of communism comes with its own group of people who are more equal than others. It seems to always turn into some sort of horror story eventually, once they get free rein to implement their utopia. I would view that with a strong dose of skepticism.