One might argue that if a church trains its members to vote a certain way, they can wield significant influence, albeit indirectly. Similarly corporations have a great deal of indirect influence. True, there’s no formal, legal coupling, but perhaps there is merit to challenging concentrations of power, no matter how informal or indirect?

That can be said for literally any sort of organization of people

Church is different in that (as far as internal logic goes) adherence/non-adherence to their dictates involves a credible claim to the fate of your immortal soul. It's a little bit different from other book clubs.

That’s a pretty arbitrary distinction. The average union worker finds not being able to put food on the table an equally terrifying prospect as eternal damnation.

I don't know if that's self-evident, given that the average non-union worker votes quite often for the union-busters who promise to enshrine their sanctifying beliefs in law.

Church has much more members than other organizations.

In many countries, trade unions have more members than churches, and they donate, endorse, and directly outright tell their members who do vote for, unlike most churches, who do not donate, do not endorse, and in almost all cases, do not outright tell members who to vote for.

Are you referring to the Catholic church? Most religious organizations are relatively small and independent from one another.

But most Evangelical churches vote as a block.

LOL only if you group together every organized religion as "Church" and compare it to other, individual organizations.

Most organisations don't threaten you with eternal torture to coerce you though...

This is a little tricky. A church cannot voice a political opinion, if they want to keep their 501 tax exemption. They can voice their opinions (1st amendment) but they'd give up those specific tax exemptions and would need to move to a different charity type. (This is why the ACLU has two organizations, one you can donate to with tax exemption and the other you cannot because they lobby the government).

The way around this seems to be for politicians to voice religious opinions, which makes some things that churches say de facto political opinions.

Seems like it should be banned on their side too, but that's a bit thorny.

Exactly. A minister (etc) can certainly say "Dancing (or whatever) is a grave sin and so we must avoid it ourselves and work to keep it out of our community." Stuff like this is protected as Free (Religious) Exercise clause and it should be.

It may so happen that only one candidate vehemently opposes dancing, and as long as they don't make that connection out loud, it stays on the level.

Is there anything wrong with a church abandoning their 501 tax exemption and voicing a political opinion?

There isn't but unfortunately what has been happening lately is churches have been voicing political opinion and not losing their 501 tax exemption because the politicians they have been speaking in favor of protect them from losing their tax status.

Nothing wrong. They'd have to pay taxes. That would be a bummer for most organizations.

Check out the history of the influence of Evangelical Christianity on American politics starting in the 1980s. Church influencing state is a contemporary phenomenon.

Money is used to buy propaganda, churches have millennia old propaganda infrastructure financed by murder and pillage, and are mostly extremely wealthy.

Any interested individual or organization can fund political propaganda, but I certainly wouldn't say the state is coupled with BLM or the NRA.

It absolutely is, in the case of the latter. BLM no, or not yet, because they are not yet an entrenched reliable source of campaign funding.

Why not? Aren't all BLM donations going to the democrats?

Maybe a notable chunk, but I'd say more accurately "to anyone but the Republicans".