The emphasis on oil may reflect a European perspective. Only 19% of US oil is imported now, and the US generates very little electricity from oil. The US is a natural gas exporter. If the US wants energy independence, it's within reach. At the other extreme, Japan imports essentially all its oil, as does the UK. Both import most of their natural gas.
In theory, the surplus profit is enjoyed by Texan oil producers, as opposed to Gulfy producers. This is increasingly dubious as the economy globalises. Gulfies own an interest in Texan oil wells. Texans own an interest in Gulf oil wells.
The Texan putting petrol in his ute... I'm not sure it matters to him that the US is a net exporter. Petrol costs him more. Oil derived products cost him more. His income does not increase.
This is not true in the reverse. In most oil producing nations, people do not pay the export price for oil. The local market price is usually lower.