What Biden has done is to cut the issuance of drilling leases to the minimum required by law, pass the Inflation Reduction Act, enact a regulation to force vehicle electrification, and similarly force fossil fuels out of most power plants.
What Biden has not done: stop issuing drilling permits or impose export restrictions on fossil fuels. The former has some serious limits because of how the courts treat the right to drill as a property right once you hold a drilling lease, and the latter is simply untested.
Nobody is suggesting that the US suddenly and instantly stop extracting, but that it be phased out in conjunction with getting rid of the need for oil in the first place.
The Saudi royal family has an alternative at this point, which is to live off their sovereign wealth fund, which owns stocks and bonds outside the country. They are also sitting on several million barrels/day in reserve extraction capacity, and could pretty easily crash oil prices if they felt like it.