Sure, it's true that religious charities are expending far more effort in helping the homeless. That is largely because we've set up our society in a way that is much more favorable to private charity (religious and otherwise) helping than to tax revenue helping. In fact, the more people donate to private charity, the less tax revenue the government gets.

I'm not disputing whether religious charities are working more effectively now. I'm disputing whether that is the right strategy. If, in 2000, you were to argue that an open-source web browser is the right thing for the world, "Most people use IE, which is closed-source" (though true) would not be not a counterargument.

It is not obvious to me that nothing can solve the problem. (To be clear, I take Jesus' words about "you will always have the poor with you" as a statement addressed to the people there at the time, not to humanity forever, but even if not, a passing comment from God still seems like an inappropriate basis for public policy in a non-theocratic society.) Even if nothing can solve the problem, it's also not obvious to me that we're anywhere close to doing as good a job as we can at reducing the problem.