Unfortunately, “getting on the step,” seems to have two distinct aspects. The first is a process or flight path maneuver to get to the best-range cruise condition. If the plane has only modest extra power for acceleration, it avoids the long time-constant of level-flight acceleration to best cruise speed. The second aspect is the idea that the cruise state reached by this maneuver, the final “step” airspeed, is extra fast; a hysteresis situation that requires a certain approach path. Many (most?) pilots don’t believe that the final speed is anything unusual.
The 1945 Boeing POH recommends climbing 500 ft. above the intended cruise altitude for each 10,000 ft. of cruise altitude. They call the best-range speed ‘the step.” It must have been the common term at that time. They never claim that it is extra fast.
In Sweeney’s description he is repeating the maneuver process again and again. So the question is not; is the final speed is extra fast? But, is the average speed of the maneuver faster? Is it faster than what? I think that it should be compared to a flight path with a constant descent rate at the best-range speed. The descent rate allows the power and fuel flow to be reduced. As Earl1947 points out, as the weight decreases you must re-trim if you keep the same power, and the speed will increase. Is this what Sweeney was doing, or was the descent and flare more emphatic?