Albert Becker annotations to the game (Wiener Schachzeitung, March 1930, pp. 76-77) may shed some light on the issue:
He critisizes 9.e3<?>, suggesting the manoeuvre <9.a3 Bxc3 10.Bxc3 0-0 11.b4 Ne4 12.Bb2> with excellent play. White hoped to execute the plan later, but he errs.
On 10.Be2, he notes that now <10.a3 Bxc3 11.Bxc3> is meat with <11...a5!> (and plater possibly <...a4>) and this prevents <12.b4? axb4 13.axb4 Nxb4! etc.>.
11...a5 receives an <!>, and is called the initation of an ingenious attack.
12.a3<?> is called a decisive mistake, since the hole on b3 is perversive of White's play. He suggests <12.Nd4!> instead.
12...a4 receives <!!> and he gives two beautiful mating lines, if White takes on b4 on moves 13 or 14.
I do not know if Becker's analysis stands up to strong modern engines, but it's certainly interesting to see which plans a strong contemporaneous master like Becker identifies in that position.
A plan involving b4 was considered fine for White. On move 9, he could have tried it. It would not have been that easy on later stages. 11.0-0-0 was not annotated, but 11.0-0 may have been better.
12...a4 fixes the ♙ formation on the queen's side, where White's ♔ tries to find shelter. There is a hole on b3, the Black ♘s can make excellent use of. Black can now try to (and does so) to open lines on the queen's side (15...d5). Another ♙-break maybe to be taken into account sometimes would be ...b5 followed by ...b4. White is prevented from improving his ♙ formation, since the Black a-♙ immobilizes both White's a- and b-♙ (if the Black a-♙ was on a5, White could under some circumstances play b3 to have both, a- and b-♙ on the same rank and answer later ...a4 with b4). There is also the psychological effect of White having to worry about Black's possibilities all the time, e. g. now it's clear that he can play ...Nb3(+) whenever he wants - when will he, and which ♘, etc.? The tactical refutation of taking the ♗b4 is also beautiful - for sure, it was also immune after, say, 12...Bb7 (13.axb4 Nxb4). The point is not so much that only 12...a4 kept it immune, since other moves did so also, but the tactics themselves. After all, Black could have overlooked it and opted for 12...Bxc3 instead (although the move does not look bad to me, but the (hidden) tactics are delightful and in this case, they were gone).