Research shows that the overwhelming majority of stocks perform better after splits.

Specifically about this point: it's an inverse causation. It's not the split that makes a stock perform better.

Stocks that are already performing well are exactly the ones that do splits, because the price of the stock has increased too much in the past and, thus, it's becoming too expensive for retail, small-investors to buy just a few of the shares. They do the split to avoid this problem.