I cannot decide whether you lack comprehension of your own native language, or whether you are deliberately obtuse. Or maybe you believe that North America's civilization, which I do not dispute, means that your property is magically safe because the people around you are a different breed from the ones populating the rest of the world.
Let me recap.
You said: I don't see how that is different from what happens on earth (aside from the space fairy dust). Whoever digs the hole generally owns the minerals extracted.
This is completely incorrect everywhere I have been, and that certainly includes the United States, where I currently work. (BTW, the countries you call 'tiny' include six of the eight largest economies in the world)
In general, oil, gas and minerals in the US belong not to whoever digs the hole, but to whoever owns the land directly above them. That is different from most other countries in the world, where they belong to the State, period. Even in the US, the resource rights can be separated from surface ownership by an explicit deed, and there are provisions according to which land owners can be forced to sell their rights, even if they are already exploiting the resources, or even if the extraction of the resources will detrimentally affect their use of their property.
Familiarize yourself with the laws of your own country! They vary from state to state, but they have a few things in common. The most important thing, of which you are clearly unaware, is that you own fuck all. The deeds, titles, etc. which allow you to use land or resources are granted by the State, and the State can unilaterally break the contract if it deems it necessary. People living on lands needed for malls, people farming above oil deposits, people raising livestock on 'frackable' terrain... those have all learned exactly how much their deeds and titles are worth. Because the US is civilized, they will be reimbursed by their losses... exactly as much as those who matter think that they should be paid.
I explicitly said: the one holding a contract with the entity able to use violence to overwhelm anyone else. Who the hell do you think I was referring to? Who do you think has its monopoly on using violence enshrined in law?
Space will be no different. Resources will belong to whoever has come to an agreement with the entity that can enforce its will (project force, has monopoly on violence, blah blah blah) Right now, there is no entity that can do this in space, which means that if you could extract the resources, you could pretend you own them as long as you stay away from Earth. Once you enter the sphere of influence of various States, things will be different.