I’ve had the same thought about the movie before, except that she didn’t dump Gosling’s character (clearly he ran out on her). It makes all the stupid things the movie does to ‘disguise’ his identity work far better, and all that needed to be done was to change his name in the credits.
It would be a lot more depressing, knowing that the other guy’s family is aware that he never really loved their mother the way he did some other woman.