There are a couple of Superman stories that very effectively show why the Clark identity is necessary.
One is a four-parter from the Schwartz era,
Who Took the Super Out Of Superman, which is available on
this site. In part 3, Superman gives up being Clark for a week. Not only does this leave him with no way to relax and unwind, but he can't appear as Clark to testify against crooks that he'd captured.
The second is from the post-Byrne era. After the Kents, Lois, and Jimmy are nearly killed by an enemy who discovered Superman's secret identity, the big S is ready to give up being Clark. Lois (who knows the secret by this time) tells him she came looking for Clark, and she can't believe she hasn't found him yet. She goads Supes into going into a cafe, as Superman, just to buy a cup of coffee. Naturally, this causes a huge commotion, and Supey realizes how important being Clark is to him.