I think what they were trying to say was that, if you have a device with large peak currents, eg motors, they will lock up as soon as the battery cant handle the current peak any more.
The powersupply will supply the peak current without a problem and power the device down to a low voltage. Whereas the battery will keep dipping past cutoff voltage as the peak current hits.
Like if you had a device that tried to pull 10A from some AA's for 0.2S every second. It might stop working as soon as the batteries fresh-state ESR had disappeared with 90% capacity left in the battery.
Which is of course true. But is a moot point because any attempt to boost the battery voltage in this situation will make the problem worse by trying to draw even higher peak current.
The only thing that would extend the battery life in this situation is a capacitor/supercap to buffer the peak currents. They would be better off selling that to clip across a cell. No issues of quiescent current to deal with.
That's the battery failure mode i see often in my logitech harmony remote control. When the batteries are getting down you only get a few repeated button presses before the mcu reboots from the IR tx led current burst.