I think your position is better, because you have many possibilities here. The move b3 does a lot of things. It immediately asks black a question — he can't really afford to allow bxc4 dxc4 Bxc4 and the response b5 fails to a4 since a6 is impossible.
It's not that a kingside attack is not justified. Actually, it's in the air. With a pawn on e5 and a space advantage on the kingside, you definitely have the possibility for a kingside attack. The closed center actually helps you achieve this. But you shouldn't always confine yourself to playing on one side of the board. The move b3 will help you on both sides of the board, so it's worth playing first.
There's a good chance black will play cxb3 here, which as you point out frees up the d3 square for the bishop, and might contribute to a future attack. It also allows you to play Ba3 and possibly trade dark squared bishops, weakening the dark squares like d6 and even f6, which could be suitable for a knight. If black plays cxb3 you also get some pressure on the a-file, and the b5 square could be occupied by your knight.
Castling long seems a bit suicidal, especially after b3 cxb3 axb3, when you have an open a-file and the a6-f1 diagonal. Thus black will probably castle kingside.
While you do have a lead in development, it's true that the center is closed, which makes your lead in development a little less dangerous. Your pieces aren't on very aggressive squares right now, either. I think your biggest advantage here is actually your space advantage, with pawns on d4 and e5 that are difficult to challenge. The move b3 threatens to increase your space advantage and your piece activity, as it might give you the d3 square and/or an open a-file.
Instead of cxb3, it is worth considering the alternative Ba6. Black is threatening to trade light-squared bishops with cxb3 axb3 Bxe2. This would be undesirable, as your light-squared bishop is better than his. But maybe bxc4 Bxc4 Bxc4 dxc4 c3 is good for you. You lose your light-squared bishop, but now you're threatening Qa4+, and if black plays b5 you have a4. If black plays Qc7 then you have Nd2, and you are still threatening Qa4+. Also you have access to the e4 square.
In lines like these, your lead in development could be more useful on the queenside than the kingside (since there is immediate pawn tension on the queenside).