I mean, I would argue that this is an area ripe for discussion when drafting new governance.

I know new nations and new constitutions don't happen often, but I would think this would need some political philosophy behind it to figure out how to do it effectively and democratically, meaning that you could trust that there wasn't corruption ongoing from people who cannot be trusted in positions of power.

The real issue is put very simply by Douglas Adams:

The major problem—one of the major problems, for there are several—one of the many major problems with governing people is that of whom you get to do it; or rather of who manages to get people to let them do it to them.

To summarize: it is a well-known fact that those people who must want to rule people are, ipso facto, those least suited to do it.

To summarize the summary: anyone who is capable of getting themselves made President should on no account be allowed to do the job.

It's why some in this thread have already suggested sortition (selection at random), but I would think there are other ways around it.

For example, representatives could be being reduced to more figurehead roles where they are there to draft legislation and bring it to the floor for a vote, but at that point, it shifts to a direct vote of the people to pass legislation. Also, before the vote is taken, the draft legislation could be put online and discussed by citizens (who can log in to the wiki using their state-issued ID or their social security number) in a Wiki format, allowing individuals to have more empowerment over how legislation is drafted and the final outcome of legislation.

Who would waste time bribing a Senator if the vote and final legal wording actually relies on the people?

Anyway, just spitballing, obviously lots of issues with that back-of-the-napkin sketch.

Lots of possibilities, the issue is finding a way to implement them reasonably in any currently existing society, which will push hard to resist such changes, because the kind of people who would exploit it are already entrenched and will do to great lengths to prevent real accountability and being expected to, you know, actually do their jobs.