We're assuming the Civil War happens as in OTL but then instead of the 13th-15th Amendments the entire thing gets rewritten, right? Because that looks like a way more interesting (and plausible) scenario in terms of state formation. Presumably we see the following changes:
- Instead of the 13th Amendment, the new constitution says explicitly that all male citizens age 21 or more are eligible to vote, which would have enormous impact on the feasibility of Jim Crow.
- Right to privacy, couched in language like secrecy of the mail. It's something democrats in mid-19c Europe wrote into their constitutions, and the US would do the same.
- More explicit delineation of federal vs. state competencies. Giving the feds exclusive rights to regulate railroads is a near-certainty, and depending on whether there is a corresponding rule that roads are a state competency, this could seriously influence American transportation policy in the 20c.
- It's possible there would be no mandatory apportionment of taxes, making the 16th Amendment unnecessary. I'm less certain about this, because the GOP was the protectionist party and in the UK at least the push for income taxes came from the Liberals, while the Tories preferred to fund the government out of tariffs. But the GOP was never the do-nothing-keep-the-rotten-boroughs party that the 19c Tories were; a lot of American state formation came from the GOP, not the Democrats.
- The amendment process gets easier - 2/3 of both houses, no state ratification (the GOP would definitely be disempowering the states in advance of readmitting the South).
Re 21c contentious issues, I imagine there's no real change to either gun rights (that comma in the 2nd Amendment probably gets deleted, though) or citizenship (the people who wrote the 14th Amendment knew they were enshrining second-generation immigrant citizenship and were for that and against the Know-Nothings). Abortion rights get read politically the same way they are today. Whatever language the 1860s' Republicans put in about free enterprise does not change labor vs. business relationships one bit.