That's precisely how ICANN was set up, as a public benefit corporation with articles of incorporation that required it to promote "the global public interest in the operational stability of the Internet." But if you control the board, in the long term you can accomplish pretty much whatever you want. Relying on regulators to give such an organization sufficient attention isn't a good strategy; their attention is spread thin across thousands of organizations, and of course they lack domain knowledge. Though, you can leverage the threat of regulators to impede the board, much like the EFF did as a vigilant third party.

Ultimately what needs to happen is for ICANN board members to be replaced with better people; perhaps people nominated by the EFF and similar organizations. Hopefully that will happen. Moving responsibilities to the EFF just shuffles labels around. The same thing that happened to ICANN could happen to EFF, it just hasn't happened yet because EFF isn't holding billions in assets.