Good analysis. The stated motivations and the true motivations rarely coincide in politics. My read of the other side of this is that the Republicans also like pushing the abortion angle to fire up the base who have dreamed of repealing Roe v Wade for decade. But they know that won't go anywhere for practical purposes, so they too enjoy distorting motivations.
You're right, both sides play these cards to fire up their bases. Both sides already have their huge multi-zillion dollar money brokers, but they still need to fire up their bases to reap even more of the lil' people's cash and pander for votes. Everyone is shooting to get that significant 'middle of the road' type voter, so they push what they decide may work to get then to lean one way or the other, and pretend the rest doesn't exist. In short, politics is bullshit. As with everything, it's all about the power and the money. I doubt most career politicians have any real core values. Term limits could've been a possible solution, but they killed that as quickly as it became a thing.
Few even brought it up, but the whole Kavanaugh thing was as much about this as anything else. They just used the abortion thing because it would stir up the emotions of feminist women ... but the anti-gun crowd doesn't have as much 'curb appeal' to the leftist base, so they never pushed that up front.