You nailed it. I'll add a couple of cents worth.
The best way for progressives to seize control of state legislatures is to enact something like a WPA 2.0, to help rural (and urban!) voters get back to work. Once progressive policies actually impact people on the ground, they'll stop voting against their self interest.
Rural voters don't want to work in Amazon warehouses or in call centers. Many of them want to work with their hands, or renew their agriculture, or help resuscitate their small and medium sized towns.
The infrastructure bill is a start, but a real solution must involve jobs that can lead people back to holding their heads up high.