I believe Prie's reasoning goes roughly like this:
1) 2.c4 is White's best move but some lines demand quite a lot of study.
2) Systems with an early bishop development to f4 or g5 fail to gain an advantage and possibly even to achieve equality because of lines with an early ...c5 and ...Qb6.
3) 2.a3 to some extent discourages 2...c5 and a3 is a useful 'extra' move against most reversed Queen's Pawn openings.
4) a3 is useful in a few of White's normal Queen's Gambit lines, so White may well consider a delayed c4 should Black play too passively.
5) The possible switching between black and white strategies and speculating in which lines a3 is useful, irrelevant or even damaging can be quite confusing.
In short the Prie System may not lead to any objective advantage to White but it leads to 'normal looking' positions where the strongest player is likely to win. Here is a game where Prie definitely was the stronger:
Prie - D.Adams, British Cht 2005
1.d4 d5 2.a3
This is Prie's preferred move-order. He also considers 1...e6 2.a3 a valid try but against 1...Nf6 he prefers 2.Nf3 because of 2.a3?! g6.
2...c5 definitely is risky in view of 3.dxc5 when White may actually try to hang on to the pawn.
3.Nf3 Nf6 4.Bg5 Be7 5.e3 h6 6.Bf4 0–0 7.Nbd2 b6 8.Ne5 Bb7 9.Qf3 c5 10.c3 (Dia)
Now it looks like a London system where it's not easy for Black to make use of his extra tempo.
10...a5 11.h4 Nfd7 12.Bd3 Nc6 13.Rh3! Ndxe5 14.dxe5 Qc7 15.Qh5
Most likely White is winning already.
15...Ba6 16.Bc2 Rfd8 17.Rg3
Now it's definitely decided.
17...Bf8 18.Bxh6 Qxe5 19.Bg5 f5 20.Nf3 Qd6 21.Bxd8 Nxd8 22.Ng5 Be7 23.Qh7+ 1–0
The London is such an easy game to play!