I think the power fans have is an overstatement. No way, do fans who believe that playing well is more important than trophies, have any bearing on the culture of the football club and on the pitch.
I take a lot of offense at the original post. Go support West Ham?
A lot of the mentality of fans are molded partly by Spurs performances over the decades. No one supported spurs because of the League cup they won over a decade ago, the FA Cup in 91, or the 80s. Venturing a guess, it was about the players and the kind of football they played.
The team needs to start challenging and that is only done by the ambitions of the players, chairman, and the manager. I think we already are. 2 finals, a few semifinals, and a title run.
Personally, if this poll was done 2/3/4 years ago, I am quite sure the results will be skewed differently.
I have voted for trophies this time.