... Contributions to political campaign funds or public statements of position (verbal or written) made on behalf of the organization in favor of or in opposition to any candidate for public office clearly violate the prohibition against political campaign activity. Violating this prohibition may result in denial or revocation of tax-exempt status and the imposition of certain excise taxes...

https://www.irs.gov/charities-non-profits/charitable-organizations/restriction-of-political-campaign-intervention-by-section-501c3-tax-exempt-organizations

politics from the pulpit lost theirs?

That's what matters (with precedent as an overlay).

Can my church denounce policies by a president? (Assuming I was actually in a church.)

Does that involve "private benefit/inurement, lobbying, political campaign activity, unrelated business income (UBI), annual reporting obligation, and operation in accordance with stated exempt purpose"?

In the absence of a campaign, no. And even then, probably still no. Even if it were during a campaign, an organization can still advocate for a cause: You're a 501(c)(3) that's advocated for carbon reduction for 5 years and a now that a candidate's in favor of it, advocate on. Just don't endorse and say vote for Candidate A; let that hang in the air. There's legislature pending? Sure. Just don't contact your legislator and, you know, actually lobby.

Abortion? Keeping out of our kitchens and garages--or uterus or bedrooms--and not push to ban something by act or encumbrance? Sure, it's okay to say, "Mind your own damn business!"

A lot of organizations have GOTV drives--targeting their members, who share like views (presumably). And it's okay to say to vote for the person--whoever that may be (wink-wink/nudge-nudge)--that you think best supports your values.

Some confuse advocacy with lobbying, including trying to change public opinion. This strikes me as mission-creep for the purpose of trying to silence others: If the majority view is X, it's easy to say that not-X is lobbying and to be banned. Rights protect minorities.

If a 501(c)(3) fails because of lobbying, it can become a 501(c)(4)

Per the IRS - In general, no organization may qualify for section 501(c)(3) status if a substantial part of its activities is attempting to influence legislation (commonly known as lobbying). A 501(c)(3) organization may engage in some lobbying, but too much lobbying activity risks loss of tax-exempt status.

Per Surfered - There is a lack of enforcement in this area.