"But in practice: how would we even write a law that hampers this behavior from Congressmen? Who will keep track of which stocks Congressmen are trading, and which committees they are a part of?"

That's pretty easy, actually.

Congresscritters are Federal employees. They already fill out lots of forms for salary and healthcare and expense reimbursement... so you add:

1. a Form 3 equivalent to register all the interests that they hold on day one, with a list of brokerages that they have accounts at. An accurate form would be a requirement to collect any salary or reimbursement.

2. an authorization sending copies of all trades to an forensic accounting office, perhaps under the auspices of the IRS. Again, don't fill this out means not collecting your salary.

3. a warning that all trades while you are a Congresscritter must be made with a 10b5-1 plan or be pre-emptively considered insider trading

There we go. While we're at it, extend this to all the positions appointed by the executive branch and confirmed by the Senate.