The point of e3 isn't always to play Nge2. The Knight can also go to f3.
The point behind closing in the Bishop early on is stability. By pulling it out early, like in the leningrad variation (4.Bg5), the dark squares on the queenside can be weak.
Also, White is anticipating getting the Bishop pair. Instead of pulling it out early, which will often lead to its trade, like h6, g5, and Nh5 followed by Nxg3, White is going for the long game, figuring either Black will play for ...c5 and ...cxd4, where White takes with the e-pawn, opening up the Bishop, or else White plays for e4 to open it.
4.e3 Bxc3+? Is ineffective. Yes, the pawns are doubled, but White has Ba3 and infiltration on the dark squares. Other than the Hubner Variation for very concrete reasons (1.d4 Nf6 2.c4 e6 3.Nc3 Bb4 4.e3 c5 5.Bd3 Nc6 6.Nf3 Bxc3+ 7.bxc3 d6), Black should wait for a3 before taking on c3. There is more to the Nimzo than just the doubled c-pawns. The Bishop pair plays a major role. So much so that even 4.a3 is a line.
I have a question regarding the Nimzo I hope some advanced players can answer. Why does white generally keep his dark squared bishop behind the pawn chain? Is it always for concrete reasons?
For example, after d4, nf6, c4, e6, nc3, bb4, why is e3 so much more popular than say, bf5. I understand that the point of immediately playing e3 is for white to bring his knight to e2, as to back up the knight on c3. However, why doesn't black just capture the knight c3 immediately after e3 and get what he wants? Even in many qc2 lines, white still plays e3 before bringing out his dark squared bishop. What am I missing?
Edit: I understand I accidentally spelt "bishop" wrong in the title. Not sure how that happened