When I played the Accelerated Dragon and had to meet the Classical Variation, my aim was simple: to sac the exchange on c3 and get pawn e4 in return.
1.e4 c5 2.Nf3 Nc6 3.d4 cxd4 4.Nxd4 g6 5.Nc3 (imprecise; I now think 5.Be2 Bg7 6.Nb3 Nf6 7.Nc3 better, but have never met it) Bg7 6.Be3 (6.Nb3 Bxc3+!?) Nf6 7.Be2 0-0 8.Nb3 (or Black will play ...d5, which is one of his main goals, even if this involves a loss of tempo with d7-d6-d5) d6 (here is the transposition) 9.f4 Be6 10.0-0 (10.g4 Rc8 11.f5 Bxb3 12.axb3 d5) Rc8 11.g4 (11.f5 Bd7 12.g4 Ne5 13.Bd4 Rxc3! or 11.Bf3 Na5 12.Nxa5 Qxa5 13.Rf2 Rxc3!) Na5 12.f5 (12.g5 Rxc3!) Bc4 13.Nxa5 Bxe2 14.Qxe2 Qxa5 15.g5 Rxc3 with equality at least for Black.
This is reason for me to believe, that 9.Be3 in the Classical Dragon is imprecise. Karpov preferred 9.Bg5, but it looks like a6 idea 10...b5 gives enough counterplay.
After I had abandoned the Accelareated Dragon 9.Re1 gaines some popularity. I don't think this is really dangerous, but also don't know what Black's best plan is.
Remains 9.Kh1, when Be6 (a6 10.f4) is best: 10.f4 a5 11.a4 Qb6! Van den Berg-Larsen, Beverwijk 1959. White should play 10.Bg5 and Black still has to prove equality.
All in all Black tries to initiate counterplay on the queen's wing indeed. The reason is simple: the c-file and the dragon-diagonal intersect on c3. If possible, Black plays ...d5 to neutralize White's centre-control.
The question is, if Black's counterplay is enough if White choses 9.Kh1 Be6 10.Bg5. Quite double-edged is the insertion of x...a5 y.a4. Sure b4 is a nice square for the black knight, but so is b5 for the white one. A serious objection is, that Black cannot use a5 anymore. A logical plan is to leave the pawn on a7 and transfer the knight-c6 to c4.
I hope this helps a bit.