It's not just plausible, it's what generally happens. The government sells licences to companies for access, or levies a tax of some sort on the extracted product.It's funny let's say hypothetically an asteroid falls down on Earth loaded with expensive resources that dwarf even the debt by a factor of 10. To lessen other variables let's say it fell on a state and no one died, and to make this even more unrealistic, and simple let's say everyone agreed it belongs to the federal government. I wouldn't be the least bit surprised if Biden straight-up just gave it to the highest US bidder for pennies on the dollar with Black Rock and Co. coming in and lobbying them when in reality the best thing to do would be to charge for it at semi market rates, pay off the debt, make SS and the Medicare plus Medicaid programs solvent, do lots of future goods spending aka infrastructure spending with high-speed rail, walkable cities for those that want it, housing, and etc.
Don't for a second think this isn't plausible.
For a country the size of the USA this can be limited: Saudi Arabia or Norway can make a fortune for the government because they have small populations and so correspondingly modest government budgets. $1 trillion worth of oil offers a country that economic size vastly more options than it offers a country the size of the USA.
Nevertheless, there is a concern about countries using that wealth in what may be relatively poor ways. The UK used its 80s oil bonanza for tax cuts, which was great at the time for people who had more cash in their pocket, but now is often viewed as a huge missed opportunity. It's certainly a concern that the USA would let a lot of the potential income slip through its fingers to give easy profits to shareholders.