Are you sure this is the dominant factor rather than one angle among many that they can use to gain political power?

If you take pretty much any issue, you'll find a divide across political parties. Climate change? If you think it's a problem, you're a Democrat. If you think not, Republican. Gun control? If you think it's a problem, you're a Democrat, if not, Republican. Think it's important to foster personal responsibility? You're a Republican; if you think everyone is always trying their best regardless you're a Democrat. Government expenditure? Ha ha, trick question! Both spend like crazy and criticize the other. Give lots of lip service, at least, to "black and brown" Americans? Democrat. So...if you're a Republican....

Care about the issues of the common American? You're a...wait. You were a Democrat, but as Democrats attracted more and more support from educated and financially well-off individuals, now Republicans are switching to that, as much as possible within the constraints of their other positions. (Trump is a populist. Tucker Carlson also.)

If you engineered a situation somehow where Republicans were guaranteed electoral domination by dumping white supremacy, I bet they'd do it. They're perfectly happy to jump on board with non-white candidates who look promising (e.g. Larry Elder was the top candidate to try to replace Gavin Newsom as governor in California). That's why I don't think it's fundamental, or really deeply in control. It's a troubling strain within the right, and one that has power far beyond its support, but even so, the power more than the whiteness is what matters. (And, for Democrats, too--to the point where they have to accuse black Republicans of advocating for white supremacy when taking a classic Republican approach to the betterment of black people--the reactions to Thomas Sowell come to mind.)