The most immediate practical consequence of a ban is that it would push a lot of talented people out of public service. You're already asking them to take a massive pay cut and deal with endless scrutiny. Now you're telling them they can't manage their own savings the way every other American can? That's a huge deterrent. The people who care most about building long-term wealth ethically will just say no thanks, and we'll end up with officials who are either wealthy enough not to care or too dependent on the system.
And here's the thing: we already have tools that work. Financial disclosures, recusal rules, and blind trusts handle the conflict-of-interest problem without a blanket ban. If someone is voting on defense contracts, they shouldn't own defense stocks. That's obvious. But banning all individual stock trading punishes everyone for the bad actions of a few.
On top of that, officials would just dump their money into ETFs and mutual funds, which makes the whole system less transparent, not more. We'd have less insight into their financial incentives, not more. The current rules, enforced properly, are better than an outright ban.