I doubt that the positive impacts of investing in publicly traded companies would be measurable or significant, and it would generally cost us returns, and that loss could be better spent. Besides alexrjl's links, here's a good video on responsible/sustainable investing. It's worth taking a look at some of the comments, too.
However, there can still be benefits to having an EA index (or ETF): it could bring attention to issues, e.g. plant-based meat, AI, etc.., and EAs may have different risk-aversions, credences in extreme events, discount rates and time horizons from the overall market (although these probably differ by cause area). There are also other tweaks that can be made to a standard market-cap weighted index to improve risk-adjusted returns, e.g. towards the other factors in this model, like value, but there are also already indices and ETFs for these factors (although usually not handling all of them together at the same time, it's up to you to combine them).
There's also this talk by John Halstead, who co-authored the paper with Hillebrandt.