Evangelical Christian nutcase groups (aka the types that push this crap) tend to play both sides like you said in the first place because it helps them get power and push this type of shit and they know they can get away with it to because they know both sides rather bitch at each other then deal with them that and they target low hanging fruits so they can then go after they real targets.
The stuff they went after on steam was said low hanging fruit the main goal is to just flat out remove all porn
Collective Shout is a feminist group, no matter how badly you want to use it to blame other groups that you hate. Every single action and statement they've made has been on the basis of feminist ideals and opposition to purported presence of sexual objectification of women in media. None of this is right-wing, or even evangelical christian. The entire basis you and other fear mongers claim this is the alt-right boogey man comes from how one of it's founders called herself a "pro-life feminist" 13 years ago. Abortion isn't mentioned or discussed anywhere at all in collective shout's public statements or actions.
To claim that what's happening here is some right wing movement is completely disingenuous. You're completely ignoring the last 10 years the left has spent pushing to remove any sexually titillating content from movies and videogames. (This is where the idea of "woke media" comes from.) Even the mechanisms being used to pressure payment processor executives into doing this are left wing. Rape, incest, and ** violate ESG and SRI investment policies and these NGOs making a public stink about them causes investment firms to divest based on this public pressure which hurts the payment processors stock performance.
Meanwhile what are the conservative politicians doing in the middle of this? Pushing an equal banking act and some age verification laws that are very poorly enforced. (Think carefully about whether you really want to support childhood access to porn before posting a knee-jerk reaction to this.)
If anything, these recent events with the payment processers have been bipartisan. Conservatives want to remove **, and Liberals want to remove rape content (which includes dubious consent content such as hypnosis).