So your problem is that you have, as you may have guessed, a small error in the construction of your quantum stockpile. In you current setup you have a minecart and stop next to a single stockpile from which it takes goods and into which it dumps goods. The proper arrangement uses 2 different stockpiles per quantum stockpile. They must be defined separately. One of them collects goods to be put into the stockpile and the other is dumped into and is only present to control the use of the goods after they have been dumped.
I will reproduce the diagram from the wiki and explain the parts.
The I area is a stockpile, call it stone stockpile #12 (for example). Its purpose is to gather items for the quantum stockpile.
The S is the track stop which dumps into stockpile O(south in this case). It is set to take from the I stockpile (stone stockpile #12).
The O is the output stockpile, call it stone stockpile #17. Its purpose is to store the contents of the stockpile group. Any hauling routes that take from the quantum stockpile should take from O(stone stockpile #17). O must not show up in the hauling route that defines the quantum stockpile. O must accept anything that will be put into the stockpile otherwise it won't hold on to them.
The S and O stockpiles must be different stockpiles. They should have different names and have been defined separately (just not part of the same rectangle you don't have to leave the stockpile tool between definitions, although you may).
The way it works is that dwarves see empty spaces in stockpile I and bring the items to it. Then another dwarf sees that there is a hauling route to put things from stockpile I into the minecart at S, so they do. The minecart immediately dumps anything put into it south, which, just by coincidence (read in a faux innocent tone), happens to be a stockpile that accepts those types of items. Since those items are in a stockpile now, and nothing takes from that stockpile, they will be left there until a dwarf needs one. This has also left an open space in stockpile I, which restarts the process.
Your current setup however looks like this:
Where I is your stockpile, and S is your minecart stop which dumps into I and also takes from it. I this situation the following happens. A dwarf sees something in stockpile I which is linked to stop S. So he takes that thing and puts it into the minecart, as he is supposed to. The minecart then immediately dumps it back out into the same stockpile and the pattern repeats, doing basically nothing.
To change your current setup into a quantum stockpile that will do what you want, you can do the following:
Remove the statues near the track stops so that you have space. Where the statues were create another stockpile that accepts the proper sort of goods.
Go into the hauling menu and change the route for the stop so that it takes from this new stockpile and not the old one.
Since the minecart stops weren't built to dump you will also need to rebuild them. Just deconstruct the track stop and then when placing it use the 'd' key to change the direction it dumps until it will dump into your new O stockpile.
Finally, the wiki notes in the second to last paragraph of the quantum stockpile section that they are not good for food and drink (or anything stored in barrels or bins.) So be sure to set the bin and barrel numbers to zero for your quantum stockpiles.
That should be sufficient. There are things you can do to clean up the route, like removing the push when full instruction, but they are not necessary to getting the quantum stockpile working.
The minecart based quantum stockpile works just like a normal stockpile, except that it takes advantage of the fact that minecarts will dump things in places that dwarves wouldn't. So you can imagine a minecart based hauling route that dumps into a large stockpile. In fact this may well be how they were first discovered. Someone set up such a system and then noticed that one square had a huge amount of stuff on it.
Hopefully this is somewhat useful.