I asked Jacob Hacker, a political scientist at Yale, for his assessment of the Drum-Linker arguments, and he wrote back:
"It strains credulity to argue that Democrats have been pushing culture-war issues more than Republicans. It’s mostly Republican elites who have accentuated these issues to attract more and more working-class white voters even as they pursue a plutocratic economic agenda that’s unpopular among those voters. Certainly, Biden has not focused much on cultural issues since entering office — his key agenda items are all bread-and-butter economic policies. Meanwhile, we have Republicans making critical race theory and transgender sports into big political issues (neither of which, so far as I can tell, hardly mattered to voters at all before they were elevated by right-wing media and the G.O.P.)."
Hacker provided me with a graphic of ideological trends from 1969 to 2020 in House and Senate voting by party that clearly shows much more movement to the right among Republicans than movement to the left among Democrats...
In this view, the left may start culture war conflicts, but the right is far more aggressive in politicizing them, both in legislative chambers and in political campaigns.