What evidence do we have that this argument is wrong?
The answer to both questions is contained in one simple phrase: TAX EXEMPT.
The reason religions are tax exempt is they do the work the government should be doing, but they do it better, faster, kinder, and at much less cost.
Notice the work done by religions is a a substitute for government programs and you see that the argument proposed is just confused. Religious activity in this area is a PUBLIC GOOD.
At most, we want to monitor religious activity to make sure they're not advocating criminal activity, as the Islamic madrassas do. If they do, pull their tax exemption.
But, after that, leave 'em alone, they're not just doing the Lord's work, they're doing Uncle Sam's work too.
The reason that Congress made a tax exempt status for religions is so they could control them through the veiled threat of revoking it.
If you object to their religious nature, don't accept their aid. Problem solved.