I'm not sure we would have won a lot more in terms of Premier Leagues and European titles (reaching 1 UEFA Cup final and 1 CL final in 22 years is quite poor IMO, Wenger never did that well in Europe), but we definitely would have had a more competitive side.
I actually read an interview with Dick Law and, by his own admission, the main men at the club responsible for transfers were overtly cautious - including himself. This totally risk-averse approach to signing players was not healthy.
However, Wenger did seem to earn the right to have a lot of control at the club. So I do believe he himself was also partially responsible for the type of players we signed, many of whom flopped in his latter years, and he did have a say regarding player contracts. I also believe he was too loyal to certain players that frequently underperformed (Almunia being a prime example).
Collectively, my belief is both the board and Wenger let us down.