I could just buy former index stocks that fall off the indexes, and take advantage of other such institutional edge cases that may not be efficient for pro's to capitalise on, no?

Why shouldn't it be efficient for pro's to capitalize on the effect? There are likely hedge funds out there that trade on the effect. Those hedge funds have complex computer models to predict which stocks are going to fall in the future of the index. As a result they are going to make the trade days, hours or minutes earlier than you.

The high frequency traders are going to make most of the profit that's going to be made on the effect.