I don’t think it assumes words have immutable meanings, just that they have some conventional meaning. You are proposing that we turn the debate from “Is eugenics plan x a good idea?” to “Does plan x, which fits the current conventional meaning of eugenics, sound like eugenics to you?” Unless you can unilaterally change the conventional meaning of eugenics, then for your purposes the meaning might as well be immutable—your argument will fail. And not only do people show no signs of being willing to shift the conventional meaning of eugenics in a pro-eugenics direction, but they seem very willing to shift the conventional meaning of eugenics in an anti-eugenics direction whenever anyone asks.
I think probably things inside and outside the Overton Window require different strategies. “Ballsy countersignaling” might work differently for things outside the window than for things inside of it. I agree that the politician shouldn’t call abortion murder.