Taylor said that he's never heard of anyone proposing a similar direct description of G and that to understand G one studies the representations of G.
I remember Mazur telling me this when I was a grad student. He made this point in the following way. You shouldn't really think of Gal(Qbar/Q) as a group which has elements, but as a "group up to conjugacy" -- thus, the aspects of Galois groups that really make sense to think about are the conjugacy-invariant things: conjugacy classes (like Frobenii) and representations.
To unpack this a bit more: a Galois group is a fundamental group. But to talk about a fundamental group (as opposed to a groupoid) you need to choose a basepoint. To talk about an absolute Galois group you also need to choose a basepoint, which is to say an algebraic closure Qbar/Q. (So just as one should talk about pi_1(X,*) rather than pi_1(X), one should talk about Gal(Qbar/Q) rather than Gal(Q).) But a basepoint you can just draw with a pencil. A Galois closure of Q is not so easy.