Well, sure! People may not know you by name, but there you are, up on that glowing box or screen, hanging from the tail of a helicopter! Millions of people will see you, point, gasp, and say, "No amount of money could get me to do that."
Of course some stunt performers have achieved fame, but are more of the daredevil variety—performing stunts for the stunts' sake, rather than as part of something bigger. The most famous of these was Evil Knievel. Surprisingly, he wasn't all that evil.