I'm just now learning the book so there is much I don't yet know about but when playing black in the old indian if white replies to ....a6 with a4 then the double step ...a5 is often a good reply. Seems to me that can apply when playing it as white reversed.
1.a3!!
But even if you play 1.a3 and the purpose is to just hoping your opponent will play Bc5 in the future, what if they just put the dark squared bishop on e7? you just lost a tempo...

But even if you play 1.a3 and the purpose is to just hoping your opponent will play Bc5 in the future, what if they just put the dark squared bishop on e7? you just lost a tempo...
Wrong. Once again, b4 is not linked to a3, or at least is a minor idea.
For example, let's have a quick look on Duda's game :
Here a3 was a useful move, preventing Na5 where Bc4 could escape to a2.
It just transposed into english reversed, like almost always with systems like 1.a3, which is (at least in this game) not a tempo lost.
I have the book. I like it. I'm tired of having to use a bunch of books with a mountain of plans and and ocean of theory to deal with. I'm going to play my entire repertoire from that one book. It will take time adjusting to playing different patterns than I'm used to.
I am the opposite and can say for theory heavy players, it's very possible to goad someone who plays 1.a3 into mainlines where ratings don't matter so much as precise moves so be careful
The book doesn't use 1 a3. It uses 1 e4 and heads for the setup.
Some of the sharpest positions in all of chess involve white having early pawns on e4 and a3
When white the pawn usually goes to d3 on the second move, like it does to d6 in the Old Indian and Hanham Philidor when black. White is going for that setup reversed.