You could play h3 just to keep the bishop, but it really just loses a valuable tempo, thus there is no significant advantage on purting the bishop there, although its fully playable.
Why is the main line e3?

You could play h3 just to keep the bishop, but it really just loses a valuable tempo, thus there is no significant advantage on purting the bishop there, although its fully playable.


Black could probably play ...dxc4 at some point if White try's to bring out the Bishop to soon.
Plus, 4.e3 gives alot of options for Black, going for sharp or more positional game.

Bf4 (or an eventual Bg5) are moves. e3 with the bishop still on c1 is also a move. Nothing wrong with either way of playing.
The "get your bishop outside the pawn chain" is useful to beginners who unnecessarily block their own pieces all the time. It's not always bad to do this though. As black playing against Bf4 you can sometimes make good use of the dark squares white has abandon on the queenside.
And e.g. on f4 it's easy development, but it's not coordinated with anything. It's a bit what I call an empty diagonal for now.

SnowRealms is right. In a perfect world white would like to have the bishop outside the pawn chain but if black is allowed to play dxc4 there will be certain complications. That doesn't mean that white must protect the pawn but he has to prepared for some pretty sharp lines if black is allowed to take so many players consider it simpler to just play e3.
By the way, I think it's wrong to say that 4. e3 is the mainline after 1. d4 d5 2. c4 e6 3. Nc3 c6. It's A mainline but white has other popular moves in e4 and Nf3.
Why not first get the bishop out and then play e3? It blocks in the bishop.