Personally I've never understood the role of fighters. When you have that much advanced weaponry and targeting power it makes sense to slap all your systems onto a platform as large as possible. That means better endurance, payload, and survivability.
The only role I can see a fighter succeeding at is hit-and-fade attacks on relatively unprotected targets. And even then, thats not beyond the capability of a frigate or escort acting independently. So why require the use of a carrier just for that?
Before running away with imagination, first consider real-life history.
real-life example #1: Pearl Harbor, December 7, 1941
real-life example #2: Battle of Midway, June 4, 1942
real-life example #3: Falklands War, HMS Sheffield sunk, May 10, 1982
real-life example #4: USS Stark Incident, May 17, 1987
In all of these examples, the airplane wins over the ship. The WWII examples are noteworthy for being strategically decisive.
The Battle of Midway is particularly interesting. In their attack on the Japanese carriers, the initial waves of American planes suffered all but total annihilation, but their sacrifice opened the way for a later wave of dive bombers to take out three Japanese carriers in about ten minutes. Those carriers never recovered from that blow and were scuttled. There is a strong case to be made that a few American dive bombers in that moment won the war in the Pacific for the Americans. All the other planes and ships participating in the battle contributed to creating that single moment of vulnerability, in which the dive bombers attacked and won. This was not specifically planned to happen this way, but it was implicitly recognized as a possible outcome when the aircraft were launched, one possibility among many worth trying to achieve and together representing a reasonably possible slice of the pie of possibilities, reasonably possible especially considering the surprise factor and the fog of war in effect.
Today, multi-role planes, such as the F/A-18, blend fighter and bomber roles into a single aircraft.
Many fighters attacking together offer more targets, and can attack independently and simultaneously along multiple trajectories and under multiple scenarios by multiple means. Furthermore, a single fighter is all it takes to defeat an unprotected target. In the Battle of Midway, in the way they died, the preceding waves of American planes made the Japanese carriers wide open vulnerable, ripe, and effectively unprotected, to the wave of dive bombers that destroyed them. They did this by pulling the Japanese fighters out of position and using up their ammunition, and causing confusion and false starts which, due to a failure to observe strict discipline, led to mismanaged ordinance on the carriers, resulting in an explosive "tinder box" effect.
Therefore, having one fighter in a critical position is the cheapest way to exploit a vulnerability. This is really the reason why fighters are important, since it maximizes the number of vulnerabilities that can possibly be simultaneously exploited, for a fixed construction cost. The deployment of many fighters at once is also flexible and in principle highly adaptive.
When real vulnerabilities are potentially present, many fighters attacking simultaneously can make effective defense difficult to plan. In and of itself, the opportunity cost, which is inherent in choosing any precise defense posture over some other, creates in theory opportunities for one fighter to succeed when many attack together.
Note the cost ratio in real-world example #3. How big the loss for the British, versus how little for the Argentinians [for that single incident].