I can see perhaps the Chinese and maybe the Russians looking to own asteroids, but that would require they pull out of several international treaties (not that big of a deal). For the West, their model has always been government-backed private monopolies (Hudson's Bay Company, East India Company, United Fruit, modern oil companies, etc). The American government would be more than happy to let Planetary resources do their own exploitation, while throwing its military weight behind them "to protect American interests". I'd be surprised if Europe makes much of an effort.
Given the ever growing need for money I can bet a tax would be put in place to where it might not even be profitable after all the regulation and such got put into position.
Think about it. They can lay levies for everything from managing the movement of objects in space to the environmental damage and insurance against such for returning materials gained in space. Then comes the whole "it belongs to all and none" type spiel that will surely follow.
You've outlined why the Secretary General of the UN might like this idea, but why on Earth would any of the space capable nations go along with this plan?
