Yes! I find it wastefully time-consuming to have to spend so much time dragging loot from inventory to storage, one-at-a-time. This is one of those game mechanics that doesn't really have any reason to not be made quicker and easier!
Additions to the suggestion:
1. Have more than one auto-drag option. It's helpful for different reasons since I don't always move all of my loot into existing stacks, especially when I'm hunting for specific loot. This is where you could have a certain key binding for sending things to storage, an open container, or between containers.
2. Allow the auto-drag key binding to be customizable. I should be able to hold shift and click an item/stack and have it go to the top section and ctrl+click to have it go to the bottom section while another player has it the opposite or something else entirely.
3. Have another option to auto-drag to the first-available quick bar slot. I usually keep them all full except when I'm replacing a dead item, so shift+ctrl+click would be a handy way to replace a dead HK in the middle of a fight.