Sounds like a proven concept, why do we still burn oil?We don't. The last oil powered carrier (the USS John F. Kennedy, CVA-67) was retired in March of 2007.
However, smaller ships have been deemed not suitable for nuclear power due to costs - no one wants half (or more) of the cost of the ship to be the power plant. That may change as oil becomes more expensive.
Actually, the problem with nuclear power for smaller ships is power/weight ratio - smaller ships (escorts, destroyers, frigates, cruisers, etc.) generally need to be capable of higher speeds than nuclear power can deliver for their weight. This gave rise to the Spruance and Burke class destroyers with their gas turbine power plants. (A Burke class has four gas turbine engines, each of which delivers some 27,000 shaft horsepower).
Big "E" was a unique beast, and was capable of legendary speed and endurance, but later carriers are, ahem, somewhat less capable? (Cost being the driving factor there).
I served aboard Enterprise my last few months in the Navy (yard duty for an overhaul), and though I didn't like her nearly as much as the Sturgeon-class attack submarine I served on, she was a good ship with a good crew. Yup, I'll miss her, and I'm wondering why she's being scrapped instead of made into a museum. She certainly deserves better after serving her nation so long and so well.