When Iron Man created the Illuminati, many people opposed of the group, and rightly so. Most members were pretentious idiots. Iron Man has made tons of bad decisions. Mr. Fantastic is known to lack common sense. Namor is constantly invading the surface. Professor X is a manipulative jerk. So, if the Illuminati was elected, who would be the members...
Being the director of SHIELD, Nick has the experience to do things the Illuminati should do.
Cap represents America in the best way possible. He's patriotic, good-hearted, and a veteran.
Even though the original Illuminati based in Wakanda, T'Challa rejected them. With the elected team, he would gladly assist.
Black Bolt would stay a member to represent the Inhumans.
Storm would represent the mutant population. She is a better person than Professor X and Cyclops. She is compassionate and reasonable while Chuck X is manipulative and Cyclops has his "daddy issues". Also, she happens to be married to Black Panther.
Namorita, unlike her father has had a better experience with the surface world. She would represent Atlantis in more diplomatic efforts.
Dr. Strange would stay a member because of his mystic knowledge.
Carol Danvers would represent SWORD. She also has the ability to take on aliens.
The Invisible Woman
Sue Storm may not be as smart as her husband, but she has a lot more sense. She would represent the general public (the 4 are like celebrities).