The rights granted by the act would be effective only between US companies, which the lawmakers do have jurisdiction over. So two different US companies go to the same asteroid, whoever gets there first gets extraction rights. Any non-US entity goes to that asteroid later, they can do what they want regardless of this law. When that starts becoming an issue then we'll see work start on a treaty (or amendment to OST) to govern this sort of thing internationally.
This bill is the first word in the process, not the last. You have to start somewhere.