If the battery needs to be discharged to reach storage mode then no, it won't be any quicker in either setting. Storage mode just tells the charger to put the cells at storage voltage, if the cells are over that voltage it will discharge, and if they are below it will charge.
I spend a lot of time flying RC planes, those drain a battery in 5 minutes or less so we take stacks of packs with us. If I'm not going to be flying for a while I'll put them all in storage mode and cycle through them in my office. If they are all drained, I can get through 15-20 packs on a dual charger during the day no problem. If they are charged and need to discharge there's no way I can get through the pile of them in a day.