IMHO the simple change that would have the biggest effect on the American political system would be to require Congresspeople to live full-time in their districts and conduct all official business over videoconference and e-mail. Lots of behavioral science has shown that the biggest generator of trust and allegiance is physical proximity and face-to-face interactions. Make all reps have their face to face interactions with their constituents and maybe they will actually start representing their constituents. It also makes lobbying a lot less economical (instead of hiring one lobbyist that can have lunch with 435 representatives, you would need 435 lobbyists, or at least 435 plane trips) and gerrymandering a bit less practical (there's a decent chance the rep would no longer live in the district and be forced to give up their seat).

That and ensuring a bidirectional feedback mechanism between the executive and legislative branch, so that laws that aren't enforced by an administration fall off the books, and presidents that don't enforce the laws lose their job. Right now, the legal corpus of the U.S. is a constantly-accreting body, which means that no matter what the President wants to do, they can find some law somewhere to justify it, and then anything they don't want to do, they just say "We don't have the resources to enforce this". This gives the President all the power. They should be a servant to the law, not its arbiter.