As long-time readers know, I love the founding ideals of the United States. Quite a lot. I have some procedural concerns, and obviously my federally-minimalist policy preferences are unlikely to be made law – but on the whole, I might be one of the more vocal supporters of what we've got going on. America's just swell, if I do say so.
I'd say there are a couple miscalculations – not really problems per se, but factual discrepancies unforeseen by the founders, things we might want to change about the system, if we were going to start over.
Most obviously: the size of Congress was essentially chosen based on the number of people who could fit in a room. We have mega-stadiums now, and I'm a big fan of fine-grained representation, so I feel we should have bumped that number up quite a lot as the country got bigger. Having ten thousand members of congress might be good, at that point you really are getting representatives out of individual communities, in a more meaningful sense. You could know your representative.
I also think nobody anticipated that Congress would fail to jealously guard their own power – we see their political tribes placed above their own power, Congress gets essentially nothing done, and defers quite a bit to the executive branch. There's not much point in separation of powers if the President can declare war and make administrative rules that have the full force of law. I think part of the issue is that, Congressional representatives don't have much personal power, only group power.
And while federalism creates competing power structures (a big win for checks and balances), the massive size of states tends to homogenize them. If you had a huge number of states – so small you could very easily work in one and live in another – they could be more different, as the choices would be more available, and you wouldn't need to sacrifice e.g. a job or job market to get the government you want.
So, in the extremely unlikely off-chance we get to try a United States style experiment again, I'd recommend many thousands of Governor-Mayors meet in the capital every once in a while to vote on the federal law bills. Or, I mean, these days, just do it by mail or online. The floor debates would be washed out with thousands of people anyway, and they'd have real duties back home. Because they'd have real local power, they'd be more skeptical of enhancing federal power, and only do so when actually necessary. The bicameral system was mostly built to slow down law making and to prevent any laws being passed to disrupt the unstable equilibrium around slavery – we solved slavery, so maybe we could have just one body where you need 75% of votes to pass a law? Or 66%?
Perhaps merging local-executive and federal-legislative branches has some secret downside that can't be overcome by simply having more people sharing the power (obviously, if we didn't increase the number of them, they'd get more concentrated power, a lose on checks and balances, but if you 100x how many there are I think the trade comes out as more-distributed power). But I think this would be something worth trying, perhaps a more stable power structure than modern America, with stronger limits on the impulses of an individual, if you can just leave their town (and jurisdiction) on foot. Plus it'd make room for Amish America, which I think we all want to see, maybe even Drones-Everywhere jurisdictions, pharmaceutical wild west jurisdictions, real laboratories of democracy that we've drifted away from.