Why don't pensions just invest in index funds generally? High required rate of returns or?

There are multiple reason:

1. If you assume that P.E is uncorrelated/has a low correlation to the stock market (subject of many years of diatribes), then you decrease volatility of your portfolio by adding it.

2. Because a pension fund has a lot of years until they need start to paying out, then it is natural for it to attempt to harvest the illiquidity risk premium.

3. The (edit: removed extra words) "high required rate of return problem" is really a defined benefit problem. A DC plan can (and probably should) just be in mostly straight indices unless it's so big it can negotiate a good fee with asset managers for other classes.

They do (and will generally track the index themselves), but PE offers a higher risk/return profile and diversification.

Yes, underfunded relative to future payout promises, so higher rates of return required to remain solvent.