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.
Unfortunately, Russia (and SA) complicate the matter. Russia earns based on the price of oil and if the US stops producing it that price goes up along with it. The world still will buy Oil and Europe especially relies on US Oil at the moment as they ween themselves off Russia’s. Oil is the main economic driver of Russia, and you can’t combat that without producing MORE. SA’s also in the mix as they have no real other (major) economic sectors to support their country and they know Oil is going away. All of this plus Climate change leave no good options on the board to choose from at this moment except to promote and support green infrastructure…which Biden has done. It all sucks.
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.