Focusing on physical control is problematic here, because it ignores the important financing and financial remedy concerns that any corporation doing the drilling will have.
Title can only be granted by a legitimate existing owner. That far offshore it is essentially unowned land. Whoever gets there first and does the work gets title by right of possession and ability to keep it.
The problem is that no US court, nor foreign court where important investors and partners and parties against whom a remedy for interference might be sought, will recognize this as a basis for asserting title. Similar claims only survived when a sovereign occupied terra nullius, but that legal principle is long dead in the post-WWII environment.
And why should there be any such guarantee? And as said before, if there's a UN program in place and there's a problem, it's the US that will bear the financial cost of any law-enforcement solution of whatever sort.
This is actually quite the opposite of what happens in an international regime-the other parties courts have to hear these cases as well. So if there are parties over which a US court can't assert jurisdiction, or assets in a foreign country that should be seized to pay judgments, it's possible to do it as long as the treaty is law in those other jurisdictions. If there is no such law, you are out of luck.
This all got started way back, when the technology was first developed to harvest manganese nodules from the floor of the seabed. I disremember; 7,000 feet of water. It was U.S. private sector's R&D which made it possible. Then came the LOST folks, trying to cut in on the profits.
One powerful argument in support of how the LOST is good for American business is the uniform support for the treaty on the part of Industry associations.
The Law of the Sea Treaty doesn't give the UN anything; it's a treaty that has to be enacted into domestic law in order to be effective, and it creates a separate regulatory body for the international seabed. It's actually a pretty textbook example of an international business agreement, not much different from the Carriage of Goods by Sea or International Arbitration treaties.