Not at all. The power to tax IS the power to control (or at least influence).
There's really no such thing as government THE SAME. Even if that exists briefly, it would soon change.
The tax code is ridiculously complex SPECIFICALLY BECAUSE the government likes to play politics with it -- give tax breaks to certain kinds of companies (solar, energy, etc.), penalize certain kinds of things (luxury taxes), and allow some things to be deducted and not others.
That being the case, how long would it be before some churches got certain tax breaks (churches that do job placement or even as severe as the 3rd Methodist Church at address X -- see Jim Wright's tax breaks) and others got penalties (churches that aren't integrated, that don't allow gay marriage, etc.)? It WOULD happen.
Even if that wasn't a HUGE problem (which it would be), it could simply be a matter of the paperwork. A hostile government could make the required paperwork very onerous, selective, or invasive.
For instance, it could be so complex that any congregation would have to hire an expensive tax attorney to keep up, effectively forcing small congregations "out of business."
It could be selective in that the requirements were structured with certain denominations in mind -- the paperwork is easy for them but hugely expensive for other denominations.
And, of course, who would worry about government invasiveness? Just imagine government asking a church for a list of their parishioners, their level of donations, how many meetings they attended, etc? Could you imagine THAT being abused?
Finally, if the government is taxing something, it (of course) has to verify it. So, then you'd have the IRS showing up to audit churches. Everyone knows that our IRS is absolutely neutral, never plays favorites, treats everyone equally, and has never been used as a pawn by an administration ... Or maybe NOT!!!!!! Huge problem. This whole thing with Lois Lerner could have been harassment of conservative churches vs conservative groups.
Hunh?
Arguably, the tax-free status of churches is unconstitutional iOTL, because it give Churches special privileges. OK, so it's a bad argument, because they could incorporate as not-for profits under other rules, but still.
How does treating churches THE SAME as any other organization contravene Religious Freedom? How would it get governments involved in church affairs?
It wouldn't. Your arguments are, IMO, totally groundless.
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Speaking as a church-going tithing Christian.