Being a representative isn't supposed to be a "get rich quick" scheme. They already make a lot of money and IMO if they're going to invest in stocks they should play by the same rules as everybody else.

Edit: On second thought, considering how much they can influence stock prices, maybe they should have stricter insider information regulations. It's a huge conflict of interest.

For example will a lawmaker holding a large portion of their wealth in Apple shares want to introduce legislation that forces Apple to allow alternative app stores (costing Apple the 15%-30% cut of all the app store sales)?

Within an organization making a decision that advantages oneself is unethical since the purpose of the most organizations is not to fulfill the personal interests of its employees.

It would make some sense to convert all property holdings of a government employee to financial instruments related to the health of the government such as treasury notes. That way their personal interests are at least aligned with those of the government. To the degree that the instruments measure aspects of the government that are not totally aligned with the governed is a problem however.

But even if you're one of those proponents of insider trading, surely this is clearly different and worse than that, because prominent politicians surely have some power to influence winners and losers.

You are right that if the officials actually cared about companies, it would not make sense. I was thinking more like hedge fund-like cold speculation. As people guessed correctly, the point is to make the job very appealing but with little ties to powerful interests. Of course is not a serious scheme, but I'm amused by the concept I guess.

Not that I expect people would be comfortable paying politicians more (see how many complain about e.g. CEO pay) so I guess we are stuck with incentives for corruption instead.