Exchanges are certainly the biggest service providers in terms of volume handled in the crypto-space, but I wouldn't say they have much influence. Most of them are just complying with laws which are being enforced by various governments or organizations around the world. There are certainly ways that exchanges can function without having to ask for KYC - look at some of the good DEXs, for example. But centralized exchanges don't want to transition to that model because it would affect their profits. So instead they sell out their principles and sell out their customers to keep their own bottom line intact.