She’s great at that! I am honestly watching mostly for the sheer delight of seeing her play awkward so well. It’s like Christopher Reeve as überdork Clark (and Brandon Routh did a good job in Superman Returns).
The “Does Cat Grant know?” reminds me of the comic book where Lex Luthor learns that Superman is really Clark Kent (from an underling? I seem to remember him firing whoever gave him the news) and he goes off on a tirade that Superman is the most powerful being on Earth! Why would he purposely want to act weak?
We as viewers expect heroes to have Secret Identities because of the comic book mythos. Those ‘in continuity’ have no reason to expect their heroes to be two people at once. You never have people wondering who Bernie Sanders changes into, or looking for Jimmy Fallon’s Secret Identity.