I work in climate policy (in the context of electrical power but the same principles apply).
Yes, ceteris paribus, oil production is counterproductive to progress mitigating climate change. Ceteris, however, are never paribus.
Oil production is a major driver of economic activity in the United States, and has a lengthy history of receiving numerous forms of support from the Federal Government for strategic reasons. In particular, "gas pump prices," are a common political talking point as a proxy for 'energy prices recognizable to, and faced directly by, individual voters.' They are considered highly salient to voters.
Gas Prices, Like All Prices, Are Functions of Supply and Demand
When most people think of policies that reduce prices, increase of supply is commonly the go-to assumption but it is not the only means by which prices can be mitigated; reduction of demand has downward effects on price as well. Moreover, reduction of demand for fossil-fuels has the simultaneous double-benefit of applying downward pressure on gas prices and reducing gross GHG emissions.
The US Gov't Doesn't Control How Much Oil Gets Produced
The volume of oil production in the US is not something the government has meaningful direct control of, it's the result of decisions made by owners of the relevant property rights to pump the oil. Prices inform those decisions and so a successful effort to reduce demand would also trigger a concomitant reduction in oil production to prevent prices from collapsing due to oversupply.
There Are Climate Mitigation Strategies That Do Not Require We Stop Emitting GHGs
Direct removal of GHGs from the atmosphere as a discrete process allows for the mitigation of Climate Change without requiring that emission of those GHGs cease - so long as the direct removal efforts are achieving greater gross removal rates (and thus a net positive removal rate). One of the Biden Administration's stated climate goals (from the link provided in the question) is "Scale up carbon removal" which describes precisely this type of activity.
TL;DR - While oil production is the precursor to GHG emissions, it is not the direct cause, and short-run discrepancies between oil production as a leading indicator of future emissions and stated goals for climate mitigation is expectable and normal (despite being unhelpful when all is said and done).
Oil production serves as a leading indicator of GHG emissions because presumably that oil will eventually be burned in some fashion. Changes in oil production however are trailing indicators of how successful climate mitigation policies are because as demand for fossil fuels fall, producers will be unable to sell inventory as quickly and so we would expect production rates to decline. Wildly successful climate mitigation efforts would cause that decline to be quite swift and sudden, but no one that I'm aware of has ever seriously accused the United States of being wildly successful in efforts to mitigate climate change.
For thee above reasons, the levels of US oil production remaining high does not necessarily support a conclusion that the US voter takes climate change less seriously than elsewhere. (There's plenty of other evidence to support such, but this is not part of it.)