I'd love to read about the new designs.
The latter inevitably slide towards autocracy. Too much power is concentrated in one person, who is almost impossible to legally remove before their term is up, and who will happily punish dissenters within the party.
In parliamentary republics, every PM is one internal party vote away from being deposed. You tend to see less of the tail wagging the dog in them.
The US has proper separation of the executive, legislative, and judicial branches of government. The legislative has a per-state and popular representation. Which part of this is "inherently unstable"?
The only part lacking a proper proportional representation (as in a parliament).
Also, the per-state representation doesn't seem to lead to good results at all. As you said, the popular representation isn't proportional, what is a more relevant flaw than anything before this point on this comment.
And that is before you get into the details that are actually bad. It's incredible that they managed to stay stable with that electoral system, for example.
That said, looks like they will have an almost perfect opportunity to fix some of those in a few years...
The US is a known bad design, nation builders working for the United States stopped trying to use this design for new countries in the 20th century, it doesn't work. It's inherently unstable and you previously got very lucky, although you have had a civil war and numerous close calls.
It's like oh, why don't we make coal-powered cars. Well because it's a known bad idea. We actually did try that, it's a bad idea, don't do it again.