Most of the state constitutions are old. It'd be different if states were trying to add those clauses now, or actually trying to enforce them. There are lots of state laws that aren't legal but since they aren't enforced they get ignored. I'm guessing that's the situation here.
I'm not sure that the White House can do anything, I think the Supreme Court would deal with the issue(but I'm not positive).
That's about it. There's a lot of laws, clauses, and the like that are either invalidated by higher courts/newer laws or are just ignored because of their age and ridiculousness (like the many "In X it's illegal to [insert benign thing here]" stories). It also takes a ton of red tape, debate (even in a roughly ceremonial capacity, as you'd still need to bring it up at official meetings of the governing body), and such.
It's easier for them to just handle any problems with it that come up from someone trying to enforce it, as they can either point to the Constitution or laws from a higher court that invalidate it or begin work on removing an archaic ruling. It's a lot smoother than trying to systematically go through every single constitution and local law set and try to remove every single problematic or outdated law or clause.
Though as has been pointed out, it's a moot point anyway. So many of this ancient stuff has been invalidated by recent rulings or was unconstitutional from the beginning (a problem when you're making laws in a time where half the nation was still unsettled wilderness and politicians could get away with a lot more bullshit or make laws that were mostly for them and their friends' personal benefit) that it would be just a time waster to try and fix what's basically a lot of giant, gnarled typos.
