Missile boats are common concepts. Was used in 1971 war by India. The idea is to launch missiles without endangering expensive assets. Missile boats are in the range of 1000 tons and can carry missiles in the range of 10 and are manufactured for low cost. These have little air defence as that adds to expense and size.
USA uses warships mainly to project power and uses carriers extensively. In fact, the second biggest airforce in the world after USAF is US navy. Let us not take USA as a benchmark. India, Pakistan, Russia, China, France use missile boats. I am sure many other navies use them too.
AGain, you are overestimating nukes. Electronics will be fried only in case of hollywood movies, not real life. In fact, US carriers are built to be protected against direct hit of tactical nukes. Don't underestimate carrier fleets. It may be convenient to simply say that nuclear bomb will solve all problems. But, that is not the case. The technology and research actually matters.