New rule: congress is an uncompensated position. You may not take financial contributions or gifts with large monetary value within 5 years of starting or finishing a term. Any violations will be confiscated, auctioned and used to pay down the national debt. Any stock trades/investments are visible on an immediate public register.
Discussion
Problem is, hardly anyone will care about the open register, except to whale-watch and follow them.
(In AU, UK and EU you can replace "hardly anyone" with "nobody willing to speak")
Removing the loophole for insider trading would be a specific treatment, but only dismantling the power pyramid can cure it.
"When buying and selling is subject to legislation, the first things bought and sold are the legislators."
- Murray Rothbard, I think
I agree there would still be problems. But I do think if financial compensation and opportunity were severely curbed, less people would seek those positions, and very few would have the wealth to sustain them long-term. A better class of person may be able to take those positions more often.