But as long as US keeps giving itself "freebies" with every foreigner's data, I'm not trusting it. I want some well established rules for how US can access my data, and only with the agreement my government.
The US is the largest monitor of the internet because most of the traffic passes through points it can monitor. Changing this would restrict US spying, but it would make it easier for other countries to do it (consider, given the announced Olympic surveillance plans) if all the traffic went through Russia). So you'd have to simply hope traffic doesn't get routed through countries who want to spy on each-other(remember, few if any countries have legal restrictions on foreign surveillance). It seems the only real solution to prevent surveillance is end to end cryptography.
The part where it gets dicier is that many of the countries trying to get control of the internet have much poorer legal protections for freedom of speech or social acceptance of it than the US (e.g. China, Russia, Saudi Arabia, Brazil).
This might be akin to the way that RMS suggests that saying "intellectual property" (instead of "copyright", "patent", "trademark", "trade secret", and several sui generis regimes) leads to conceptual muddles. "Internet governance" may have the same problem; maybe it's a term in need of some disaggregation!
That there are, at least, cogent arguments on both sides against it suggests to me that someone's campaign somewhere is considering it.