I don't know that "inclination to judge" is a necessary component of religious charity. Religious charities beyond healthcare give out a lot of "free shit no questions asked," and usually without any attempt at proselytization. Catholic hospitals are not known for turning down patients due to moral judgements.

If there's a benefit to religious charity, it lies not in the "inclination to judge" but in maintaining a multiplicity of authorities. Combining everything under the government umbrella gives you a totalitarian society; a liberal democracy restrains the power of government while allowing for competing authorities in other spheres, like medicine, religion, academia, and the press. Giving the government the power to mandate or withhold medical treatment for political opponents (think Soviet "sluggish schizophrenia" as a diagnosis for removing dissidents from public life) seems more worrying than strippers getting free healthcare.