The problem with that is Clark doesn't have to deal with having dual identity on his own. He has many Superhero friends with dual identities that can confide in as well as his cousin who has the same powers as him and a "Super" based superhero identity. He can be himself with anyone in the Superhero community.
Also (in my opinion) Superman having to deal with the hardships of being a Superhero on his own would get so stale after a while. It was the same with Spider-Man in the 1980's. This is something that came up regularly in 1980's Spider-Man comics and it got so boring after a while.