Aren't there two main reasons?

1. You need some way of normalizing shares to produce a statistical sampling of the market, and capitalization is the obvious, existing market mechanism to do this.

2. You need to resist brute attacks on the index investment strategy. I.e. if a giant fund is known to just consider all shares equal, you could soak them by doing wild stock splits to put more of your shares on the market. Or if they consider all companies equal, you could soak them by bringing lots of empty shell companies to the market.