Hathaway

You would be mistaken about saving fuel.  Recall in our discussion here we mention the "Optimum" cruise altitude. It isn't as simple as "higher is better", the optimum altitude for fuel burn is lower than the maximum altitude you can reach, and increases as you get lighter.  When you fly above the optimum altitude, the increased AoA required to maintain altitude results in an increase in induced drag and thus increased fuel burn.
 
So in actuality, step climbing saves you MORE fuel than simply climbing as high as you can as soon as you can.
 
To be fair, it sounds like you might be misinterpreting what the pilot said and might be mixing a few different concepts together. Step climbs aren't forced by any wing issues or lack of power, but are a natural result of pursusing an optimum altitude as it changes with your aircraft's weight.