They very much give a shit about things like abortion rights and gay rights. Both are fundamental to our understanding of gender equality, and capitalism depends on the oppression of women and sharply defined gender rolls.
A post of mine from a while back:
They care about racism, sexism, and other forms of bigotry very much. You said as much yourself.
"And by having these serious differences they know they can keep us fighting between left and right while they get what they want on things that do matter to them"
It has been a tactic of the American elite since its very beginning. Our political and economic system is uniquely designed to fracture the working class, to keep us from uniting. It's why laws were passed in the early 1800s forbidding interracial communications--the poor Irish and the slaves were beginning to cause worry. It's why whites and blacks have had their hatreds stoked by their literal and financial masters against one another for the last 200 years. It's why gay marriage isn't legal everywhere yet, despite a majority of people supporting it. It's why women are still discriminated against (though there is a far, far longer history to that than just our political system, and indeed, political systems at all). Our entire political history has been carefully balanced oppression stacked on top of and woven into more carefully balanced oppression. Sometimes it worked, sometimes it didn't. So far it's worked more than it hasn't.
It is absolutely critical to keep us divided amongst ourselves. The elite care more about those issues than you might think, because without them, the people will realize their own power as one. They will become a single working class, rather than blacks and whites, men and women, straight and gay. We are a dangerous mass of humanity that would roll over them like the ocean rolls over a few pebbles.
This is why these social issues are just as important as the economic ones, the power struggles. They are utterly critical to maintaining the illusion that there is no true working class, that we are separate, different from one another. The elite care very much about maintaining that illusion.
The reason we see a split is because our two-party system has to have some differences, and equality is a rather popular demand. By having one party concede to those issues, the elite can "address" these inequalities without fundamentally changing anything. It is a deradicalizing tactic.