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TheWildEcho Secular shelters have never forced people to attend prayer or dress a certain way in order to get food and shelter. The problem with attending prayer is that not all homeless people appreciate being made to read the bible and sit through hour long prayer sessions and two it gets in the way of finding jobs when two sometimes three times per day is required. There was a good news article about either New York or California not being able to arrest homeless people if they refuse shelter but sleep on the streets instead (because many of them simply don't want to be forced to pray.)
Then you have the incidents of gay people getting kicked out for being gay in religious based charities.
Secular ones do have problems but their problems is lack of funding and people rather than any of the above.
Secular does not equal atheism so it wouldn't be an "atheist" shelter just because it's secular:
https://www.huffpost.com/entry/secularism-is-not-atheism_b_1699588https://www.jta.org/2015/10/21/lifestyle/the-only-safe-place-for-battered-orthodox-jewish-women-in-israelThe only safe place for an abused Orthodox Jewish woman is a secular homeless shelter.