We would need a Constitutional amendment to explicitly state that spending money on political campaigns doesn't count as an exercise of freedom of speech (1st Amendment). The Supreme Court Citizens United v. FEC decision was built on that legal foundation. We can argue about whether the court's interpretation was correct or politicized, and it might be reversed someday by different justices, but ultimately the only stable solution will be an amendment.

Any sort of reform that would successfully get money out of politics would require a constitutional amendment. I have little faith that the current Congress would pass any sort of amendment that would eliminate the system they benefit from unless there was at least a reasonable possibility that the states would call for an Article V convention, but the state legislatures also benefit from the current corrupt system so I don't see this happening either. You can't wholesale replace Congress either because the lack of term limits and the staggered terms means that the special interest groups that benefit from corruption will try to primary candidates before you can build up a critical mass (see what happened to Jamaal Bowman and Cori Bush).

One approach is to reduce discontinuities and spoiler-effects in voting, which in turn reduces the incentives for a politician to accept—or someone else to offer—a devil's bargain. As it stands now, there's lots of potential for: "Do me a favor and my money will push you the small distance that distinguishes total defeat from total victory."

That leads to some game theory and math, with things like some form of ranked-voting, uncapping the House of Representatives, and maybe even proportional-representation rather than lots of just-one-winner races.

It begins with hierarchy

Others have offered contrived rules for solving it through policy, but these don't account for how to get those with power over us to institute such rules to bind their own hands and then to follow the rules to their own detriment

I think some places have the government give every party equal (by what measure? I don't remember. Past votes maybe?) money for campaigning and no donations are allowed.