I think you're right about 5. e3 being the weak move. The Qc2 was championed by Capablanca I believe. He seriously disliked ruffled pawn structures and the Nimzo capture of BxN, bc saddles white with very weak doubled pawns.
But part of the idea behind Qc2 is that after 4..O-O white grabs control of the center with 5 e4!? Black is now struggling.
The pawn on d4 is going to be a white weakness for awhile and black should lash out at it if playing 4..O-O. c5 is a natural way to do that for example.
The problem with castling on move 4 against the classical variation is white has been slugging away for control of the center and castling doesn't fight back for control.
While castling as black is certainly the most common move for black the numbers don't bear out its selection. White out scores black almost 2:1 in wins and black's winning percentage is around 15%. Rather dismal.
It seems odd to me that castling would be considered normal. This isn't white playing the Ruy Lopez with a tempo up to start, the pin against black's N and heading into an open game where the rook on the open file is going to be important. The Nimzo casling against the classical system seems premature.
This is a recent correspondence chess game that I played pretty well. When my opponent mixed up two lines in the Nimzo-Indian, I formed an opening strategy around center light square control. I was able to obtain a positional advantage and used a tactical sequence to break through and win.
Also, I didn't do extensive engine analysis, so it's possible I missed things, but I don't think I blundered significantly.