Well...
(IRS) Exemption Requirements - Political Organizations
A political organization subject to section 527 is a party, committee, association, fund, or other organization (whether or not incorporated) organized and operated primarily for the purpose of directly or indirectly accepting contributions or making expenditures, or both, for an exempt function.
Exempt Function - Political Organization
The exempt function of a political organization is influencing or attempting to influence the selection, nomination, election or appointment of an individual to a federal, state, or local public office or office in a political organization. The election of Presidential or Vice-Presidential electors is also part of the exempt function of a political organization. Activities that directly or indirectly relate to or support an exempt function are exempt function activities.
So, why pick on churches for being political if there is specific category of tax exemptions for political organizations?
Further to that, would it not strike against First Amendment to suppress political communication, at least within a reasonable range? I.e. a pastor who spent all their time campaigning is not good, but one who voiced their opinion to their flock would be hard to object to. What about a pastor who campaigned very heavily for say the Democrats because of their theological beliefs about caring for the poor and the sick? What about, well might as well say it: abortion. I am firmly pro-choice but I understand entirely how many religious people object to it. They shouldn't dictate to others, true, but they can't have, political, opinion?
Last, and not directly legal, but rather political in consideration, this strikes me as a really divisive and pointless action. Things being what they are this type of measure would be highly partisan and would affect Republicans - surprise - much more than Democrats. For some strange reasons Christians in the USA believe themselves under attack by secularists despite being an unusually religious country by most rich Western metrics. This type of measure would give them more grounds than usual to complain. It would not pass and why stir this pot up?
i.e. Make no church tax exempt (not realistic politically, but I agree with the sentiment). Or leave them alone, at least for politics. Removing tax exemption for for-profit tele evangelist type outfits is a much more defensible action than policing religious political speech.