"Reversed" Openings in general
A black defense and its white mirror counterpart will often play out quite differently (compare the rather sharp Sicilian Defense and the rather quiet English Opening).
The right to move matters, in both ways. By moving, you give away information to your opponent - the set of variations you can choose from shrinks with every opening move. Early moves are usually played to either proactively increase your control over the board or reactively counter your opponent's moves (the best moves do both). This second category of moves only make sense as answer to one of the first category: If your opponent doesn't play a move that you have to react to, you'd want to play a move to advance your own plan rather than a "counter" to a plan your opponent has not committed to yet. Commitment is an important factor here. If you parry right before knowing which side your opponent will commit to, they can now simply swing left and catch you off-balance. Even if they cannot immediately strike that way, you will still have wasted time you could have used for more useful moves. The further in advance your opponent knows what setup you are heading for, the easier it is for them to screw your plan.
Most openings are a series of moves reacting to each other. Simply playing a "reversed" opening won't lead to the same result because the dynamics can be totally different depending on who strikes first.
Caro-Kann vs. 1.c3
The Caro-Kann defense strategically centers around the d5 square. Playing 1...c6 merely is a means to the end of playing 2...d5 next turn and being able to take back on d5 with a pawn if White plays exd5. Playing 1...d5 right away is certainly possible (leading to the Scandinavian Defense), but then Black concedes that he has to take back on d5 with a piece, when he will obtain a half open d- instead of c-file (generally speaking, a half-open c-file and two center pawns are more desirable than a half-open d-file and only one center pawn). Of course, White can refuse to take on d5, but that's the idea.
Now, let's have a look at the same idea, but from the white side. Sure, White can play 1.c3 to prepare 2.d4. But since Black obviously did not have time to play ...e5 yet, this is simply not necessary and White can play 1.d4 right away! Now Black won't even be able to play ...e5 with ease (the best moves: activity as well as prevention). Isn't preventing the black pawn from entering e5 not much better than preparing to attack it with 1.c3 after the fact? And remember, Black did not commit to this pawn move yet anyways and may answer 1.c3 with a completely different move, leaving White look rather silly for preparing against something that may never happen.
On the other hand, if Black does not play ...e5, White doesn't actually have to defend d4 by playing c3 ... and the c-pawn is free to be moved to c4, from where it attacks d5. If it now gets traded for Black's d-pawn, we may actually arrive at a similar situation as in the Caro-Kann exchange variation: White has pawns on both central files and a half-open c-file.
To no surprise (and in contrast to 1.c3), 1. d4 2. c4 is one of the most popular ways to open the game as White of all time. So why not go for a Queen's Gambit if you like the Caro-Kann?