I would be concerned about timing the trades of index funds, so I would propose something like you have to do a one-time rebalancing of your portfolio into index funds and it remains locked up for the entire duration of your term.

But they should definitely be able to invest in index funds at the start because you want them to have some skin in the game of the economy that they are in effect managing.

Yeah it does, because part of insider trading is simply trading on overall market beta.

Depends on the way the laws are written. Also blind trusts.