I think there's a subtle distinction between morality wars on the left and the right, at least one that I can see from my own perspective.
On the right, there's a coherent, objective moral value statement that voters want upheld, namely "babies are people too, beginning at conception, and must be treated as such by the law." It's pretty straightforward whether this value is being upheld, and there's no risk of a "purity spiral" where people propose increasingly crazy ideas to signal their loyalty to the cause. Just don't kill babies... that is all.
On the left, unless I'm mistaken, there aren't coherent, objective moral value statements at the heart of CRT, identity politics, or pro-choice arguments ("my body, my choice" isn't a well-defined philosophical statement) which does lead to a purity spiral. For example, there have been serious proposals to allow "after birth abortion" -- i.e. it's OK to kill babies after they're born - for a recent example see [0], but there are many other examples as well. TA mentions other purity spirals that occur on the left.
But regardless, yes, there are absolutely politicians on both sides of the aisle who use people's moral values as a means to power (I'm especially thinking of a recent US president, but there are many others). "You want to get rid of abortion? Vote for me!" -- but disingenuously. They don't appear to care personally about whatever it is, they posture to get power while not understanding the nuances of how to wisely implement laws that reflect the peoples' values, or even what those values are exactly in the first place.
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[0] CA AB-2223, in its original text (now amended), read "Notwithstanding any other law, a person shall not be subject to civil or criminal liability or penalty, or otherwise deprived of their rights, based on their actions or omissions with respect to their pregnancy or actual, potential, or alleged pregnancy outcome, including miscarriage, stillbirth, or abortion, or perinatal death" (emphasis mine). Note "perinatal death" means death within the first 7 days after birth (thus legalizing infanticide). This has thankfully been edited to perinatal death due to a pregnancy-related cause -- but that could still encompass scenarios like "intentional injury to the child moments before or during birth".