- no more options/shares for employees;
- significant insider buying/holding is generally considered a positive signal, alignment of incentives, etc. would need replacing with a more complicated compensation agreement linking bonuses to average share price over the year or something.
Slightly less obvious one is that you still have inside information on the index fund, not just that your company might itself be a significant constituent, but inside knowledge of the industry, what's happening to a common supplier, etc.
And by 'index fund' you probably actually mean/want 'significantly diverse pool of shares', an index can be small. Nothing (other than it being silly) stops it being just TSLA or whatever either, 'My Favourite 1' can be an index of my 1 favourite shares, and you can create a fund that tracks it. Obviously that's silly and doesn't happen to that extent, but it might if you incentivised it by making 'index fund' the rule.