However, paved runways eventually became the rule rather than the exception. Retractable landing gear made the nose-landing-gear drag penalty disappear. The nose landing gear arrangement also became of benefit during the takeoff roll of a multi-engine airplane: in the event of an engine failure, the pilot was more likely to be able to keep the airplane headed down the runway if it had a nose wheel to help resist the turning moment. In fact, it was more widely incorporated on early bombers like the B-24, B-25, B-26 etc. than fighters. (One exception was the P-38 but the configuration of the former wasn't really suitable for a tail wheel and it was also multi-engined.)
Another exception was the Bell P-39. It's unique inline arrangement of cannon, cockpit, and engine provided room for a nose gear and the need for nondisposable weight up front for balance reasons.
The Navy's preference for a tail-wheel configuration was so strong, that when it agreed to consider a carrier-based variant of the Army's new fighter, it insisted on it. The resulting Bell FL-1 became a taildragger.
For more on the brief history of the XFL-1 Aerobonita (as well as a summary of the history of the development of—and the engine manufacturer's struggle for supremacy between—the air-cooled versus liquid-cooled airplane engine and the Navy's ongoing interest in the liquid-cooled engine) see my monograph;
The Navy didn't completely forswear the nose-wheel configuration any more than they did the liquid-cooled engine. In August 1939, they went to the trouble of converting a Lockheed Junior to have a nose gear (it was fixed and the main landing gear had to be moved aft as well) and conducting successful at-sea trials aboard Lexington (CV-2).
Douglas won contracts during World War II for two single-engine carrier-based attack planes with nose landing gears, the SB2D and TB2D:
My guess is that the advantage was that heavy bombs and torpedoes didn't have to be positioned at an angle to be attached to the airplanes. A small quantity of BTDs, a development of the SB2D, were built for evaluation. Douglas quickly replaced it with an all-new design with a tail wheel, the BT2D, which became the AD Skyraider.
The Ryan FR-1 also needed a nose wheel because it had a jet engine in the tail. From time to time, it was a poster child for the benefit of not having a nose landing gear during a carrier landing (as well as proper location of the attach point for the tailhook).
Jets, of course, all but had to have a nose landing gear for various reasons. There were a few exceptions early on, even one that was carrier based, the Supermarine Attacker.
For more on twin-engine carrier-based airplanes with nose landing gears, see http://thanlont.blogspot.com/2010/11/one-if-by-land-two-if-by-sea.html