Are non-religious homeless shelters any better/worse? Obiously not all christians are saints and not everyone who calls himself a christian follows christ.
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.
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.