We're still in living memory of highly sectarian societies where, for example in Australia, a Catholic was a lower social grade than Church of England and being a Methodist was something else entirely. All of these groups voted differently, associated differently and married differently. This hung on longer in small towns in Australia but was a definite societal attribute right up until the mid-1960s where it started to break down.
We had waves of social change that barely shook that structure: Catholic churches and schools were once almost exclusively Irish, but the immigration from Italy during the 1940s/1950s changed that ethnic mixture significantly with barely a murmur to the social structure or how those people voted (predominantly aligned with the left for working class reasons). The church publically told them how to vote and they did it.
As older churches like the Methodists disappeared (or were subsumed into more liberal church traditions) and the left started embracing social reforms (no-fault divorce, womens liberation, abortion, birth control, the breakdown of racial segregation) the religious part of society bifurcated: the more religious folk went to the right and the less religious folk just started walking away from church.
That division was the end of organised religion as a driving social force. Now, the thing most christians seem to despise the most is other christians of a slightly different faith.
Most organised religions have only themselves to blame for their quickly falling church attendance for that reason.
In one respect though, the US and Australia share the history that our leaders have predominantly been protestant. I think that's far more a social class structure artifact than anything blatantly religious though.
The denomination of the leader doesn't always dictate who gets the ear of the party in our political system though - Gough Whitlam was famously coerced into including the catholic schools into our public funded school system because they were "broke" - a somewhat laughable assertion even at the time, when the Catholic church was, and is, one of the richest institutions on the planet.
The memory hole in 1984 wasn't just something Orwell made up from nothing.
Before that, the political issue churches were supporting was civil rights; a century before that, it was the abolition of slavery.
I don't think it is possible for anyone who takes positions on right or wrong to be truly apolitical: 'I avoid politics' mostly means 'I find the status quo tolerable.'
The civil rights struggle has largely been about society coming to terms with oppressed groups being worthy of having equal human rights, whether that was on the basis of race, gender, sexuality, nationality, religion, disability, whatever. Suggesting that a fetus doesn't deserve a right to live is no different than suggesting that any other marginalized, dehumanized group doesn't.
In essence, human rights for the unborn today are where human rights for Black people were in the 1850s. Hopefully someday we'll treat young life with the same respect we treat old life.
(This of course ignores the fact that the ideological basis for abortion ban is literally just stupid; it was Acquinas himself who explained how flawed the idea of "life from conception" was.)
Some undeniably do care about abortion. Some have financial ties to politics and abortion is the easiest wedge issue that they can claim to religiously justify.