You need to move the mine cart 1 tile so that it dumps the contents. It only dumps the contents when it is moved onto a tile with a track stop. This means your quantum stockpile mine cart route needs to be at minimum 2 tiles long, a starting position and the ending position with the track stop.
Each mine cart can hold 5 stone at a time, and there's some downtime where the cart is being moved back and forth. To not have any interruption when quantum stockpling multiple minecart routes can be created that are duplicates of each other. 3 minecart routes for stockpiling stone mean that 15 stone can be hauled at once, improving speed at which things are quantum stockpiled at the cost of increased labor requirements.
Your Quantum Stock Piles are set up wrong.
First construct your track stop, for this example set it to dump south. This can be change with q later if you forget or want to change direction.
Make a 1x1 stockpile adjacent south of the stop and set it for Stone only. Press q and then a and make sure it reads "accept from links only" and set it to have no bins or barrels or wheel barrows. This is your QSP
Make a 3x3 stockpile starting southwest/bottom left of your track stop (left immediately of the 1x1) and take it to the top right of your track stop, the radius is exactly 3x3. It will not over write your stop or the other stockpile if done this way. Set it to no bins or barrels. This is your Feeder Stock Pile.
Now press h, then r to create a new route. Hover over your stock and make a new stop (s I think it is) press x to delete all conditions (this is important!!!)
Now hover over your larger stock pile and press s to make a new stockpile link.
Congratulations you just made a new quantum stockpile, using just a 3x3 radius. The same size as workshop! The wheelbarrow issue is caused by allowing wheelbarrows in the quantum stock pile I think, so make sure to set those to zero.
I make my QSP setup 3x3 to appease my ocd so I can have stockpiles adjacent to my workshops. So I'll have a pile for bars next to my forge, and stone pile next to my mason.
Oh. And be careful with making multiple QSPs that accept the same items. It's likely you will have QSPs that are just empty.
EDIT: make sure to set the cart to be pushed, not guided! This will NOT work with guided Carts