didn’t just flash the image once per frame
Hmm... I'm detecting that we may be operating with different definitions - made no more clear by the fact that you are talking about projectors while I'm answering a question about cameras. To me, a frame is a single still image on a reel of film.
I think I see your point that a projector would insert more black transitions from on frame to the next so that the black transition from one frame to the next isn't as noticeable. However, I don't see the connection to original discussion about frames per second or where the film industry standard of 24fps came from.