Fame is a fickle mistress, friend. There have been many famous fighter pilots over the years, but most of them have become famous either through killing a bunch of people, getting killed or captured themselves...or becoming an astronaut. Except for Ted Williams, but he was already famous. Wait, you don't know who Ted Williams is? Outfielder for the Red Sox? One of the greatest hitters of all time? He was basically Derek Jeter with a fighter plane.
Some of the most celebrated heroes of two World Wars were fighter pilots, but with air defense systems and missiles that can travel thousands of miles and land (generally) where they're supposed to, modern warfare doesn't include the incredible frequency of air-to-air combat experienced in the early half of the last century. Think about it: The Battle of Britain was a major turning point in the Second World War, and it was fought entirely in the skies. Stuff like that just doesn't happen anymore.
A handful of astronauts—including Neil Armstrong and Buzz Aldrin, the first two men on the moon—were former fighter pilots, but it's not like they became famous for their dogfighting skills.
One of the most famous fighter pilots of the last half-century is former Presidential candidate John McCain, but even his fame came at a price; his plane was shot down in Vietnam, and he spent more than five years as a prisoner of war. If that sounds like a good way to get famous to you, we're certainly not gonna stand in your way.