I agree we should not tear down the charities first, but I don't think that we should pretend the charities are successfully solving the problem. They are making the problem less bad than it otherwise would be, and that's valuable, but they're not actually solving it. People have been saying "Private charity will solve the problem" since Judas wanted to sell that one bottle of ointment, and the poor are still with us.
The parent poster did not say that religious or secular charities could "solve the problem"; nothing can solve the problem. What the parent poster did say is that religious charities are expending far more effort in helping the homeless.
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.