I think the general premise is that prices are rising because of supply shortages and therefore we must drill at a greater rate. If that were true, there would be gas stations running out of gasoline and cars would be forming lines around those stations that had gas. I don't see that happening. The data suggests that oil coming out of the ground is matching demand for oil products.
Sure, if oil companies drilled more than could be fed to the market, prices would fall but why would oil producers oversupply? It's in the producer's interest to match demand and no more. Thus, the reason producers aren't drilling in all these other places is that they don't have to.