Perhaps, but I think that misses the point of Gunn harkening back to Silver Age comics with this movie. Puzzling out the "logic" of who knows what and why is sort of antithetical to the exercise being conducted. It's like the folks who get wrapped around the axle about the conclusion of Superman 1978 not making a lick of sense to someone with an elementary understanding of physics. Oh, Superman can turn back time by flying around the Earth backwards? That's absurd!! Yes, it is, now shut up and pass me the popcorn.
If it is necessary for the story, Superman's identity will become relevant. Otherwise, everything operates on a shrug and a hand wave, and I'm sort of fine with that. Secret identity management is a fun aspect of a character like Spiderman, for example, because his whole thing is the burden of being a gifted individual. Great power = great responsibility and all. Therefore, making Peter Parker suffer because of Spiderman is kind of baked into the text of the character.
Meanwhile, I think Gunn's approach to Superman/Clark is that neither is burdened by the other. I'm not even really sure he views it as a duality in that way. The text of the film seems to indicate that, in Gunn's view, Clark and Superman are indistinguishable from one another, or even that Clark (the human) takes primacy over Kal-El/Superman. This intentionally contrasts with the Snyderverse interpretation of DC heroes, which was much more interested in how INhuman the DC canon of heroes were, Superman most of all.