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.
That doesn’t actually work. Nuclear winter is caused by the stuff which gets mixed up with the blast. Hit Antarctica and all you get is water.
On top of that, it’s where air descends from the stratosphere, so whatever particulates you do generate probably won’t achive worldwide distribution at significant concentrations
All you get is water but it’s not just water.
Water in the admosphere is an extremely strong (but short lived) greenhouse gas.
And while it was hit, it could also be irradiated. While a nuclear blast has less radiation impact than a nuclear plant burning, throwing many nuclear bombs in one place may have other impacts. Contaminated water can be assimilated by living things. And while in the body, it can do damage.