No, it actually wouldn't — the whole power of 501(c) is that it is exemptions (and possibly affords tax-credits/writeoffs).
My sense is that if “income” is the item to be taxed, then churches will be viewed as having “income” that comes into the church via the offering plate.
I want to get rid of the income tax. I’ll agree that a flat tax is a positive first step. Then we should repeal the 16th.
Also, in the meantime, families, the same as businesses, should be able to totally write off all expenses before arriving at their actual income.