In general, you should trust what ICM Trainer says over the charts. The charts are a simplification which can be memorized, or printed out and posted by your computer, but they overlook many important features of the situation such as whether you are pushing into players who cover you by a lot.
ICM Trainer tries to calculate the game theoretically optimal play given the stacks and blinds, and the assumptions that people play push/fold, and that the ICM is accurate. The main reason to question what ICM Trainer says is if you feel you have a better read on how your opponents play, in which case you can use a tool like SNG Wizard. There are also times to modify the ICM, such as when you are about to hit the blinds, or if a player is sitting out. If those aren't issues, then try to follow ICM Trainer instead of the charts.
Weak offsuit aces are often not Nash equilibrium pushes from the CO, particularly when you push into players who cover you by a lot, who are therefore not risk-averse against you. Usually they are pushes from the BTN.