Particularly in a $2 tournament, your opponents will not play push/fold, and you need to know how to respond to them even if you will play push/fold.
When the average stack is 11 bb, this leaves open the possibility that there will be resteals instead of pushes. Many players feel uncomfortable about shoving for over 10 bb, and they will make small raises. Do you have fold equity if you shove for 11 bb total over a 3 bb raise? Should you try a stop-and-go, which means calling preflop to push on most flops?
There are a few situations in which you might not want to play push/fold even with under 12 bb stacks. However, if everyone folds in front of you, you will not lose much by playing push/fold.
Nevertheless, there is still a lot of skill left in a push/fold game. Even if you are beating your opponents, it is likely that you can still improve your assessment of their ranges and how you should respond, and that you can improve your ROI.