But I think the way our current system solves this is by forcing companies to disclose important insider information to shareholders all at once, rather than by letting insiders profit by leaking the information through their market transactions. (The one possibly being related to the other, because insiders who could legally profit by keeping information secret rather than disclose it would be incentivised to do so.) I don't think a similar restriction is feasible for prediction markets, because often the thing you're trying to predict is not something that can be legally cajoled into giving up its secrets.
Before Salman vs US (2016) it basically was, how do you think the oracle made bank?
With prediction markets, the "purpose" is information discovery, and "insider trading" actually helps (=> via information from insiders).
Disclaimer: I'm somewhat playing devils advocate here, I personally think that prediction markets are for now mostly an ineffective zero-sum game (and legalized gambling with all the drawbacks that brings).
But you don't usually buy the stocks from the company itself, do you? Unless there is some shenanigans with buyouts going on...
For that very reason, insider knowledge, and especially the ability to influence future outcomes, become the subject of heavy regulation. And, the lack of such regulation for congressional members is also why their net worth tends to skyrocket once entering office.