You had the dwarves carrying the minecarts as if they were really big buckets of lava, which was slow and took a lot of dwarfpower. Is it possible to get the dwarves to push the carts up to the stocpile via a carved track?
Sure. It requires you mining out said track, and then carving the track. At that point, you could automate the delivery with only a little more work. If you only want one or two pockets filled though, it's a whole lot less work to just have them carry 'em up, as they're moving at about 1/2 the speed they would normally carrying the minecart when it's full. I also do the same usually for carrying sand bags down to the magma forge... it's just not worth carving the track when it's just one dwarf for a month or two dragging the thing down there twice a year.
For something I need to happen three times, and only three times, per hole? It's just not worth laying the track down.
Part 6 is going to be long. I'm assembling the OMG (Overfiend Minecart Grinder) Torpedo Launcher and will be filling it with magma. A LOT of magma. The sea will feel empty because of me lot of magma. And I plan to do it quickly.
In that I'll walk through an elevator that you can use to get a lot of magma somewhere quickly, but I'd recommend you read the walkthrough that's in my signature for a massive ore dropper if you want to know the specifics of the components.
Minecarts are really confusing for me, and I've only experimented with them a little bit in much more developed forts with plentiful labor and a lower price of failure.
Same for me, at first. Then I finally knuckled down and built the monstrousity that's in my link... and realized they really weren't that tough after all, once you've experimented enough with them to get a general handle on their characteristics. They require patience, though. It's not something you just slap together. You need to have a methodical plan to use them well, always have a cart escape built in, and test the design every so often to make sure each piece works correctly so you can adjust before you've got this roller-coaster monstrousity causing issues. I don't know cart speeds like some folks do where I know what tic it will teleport on, I just know what usually works, and tweak it slightly one way or the other when it's not quite right.
Minecarts are possibly one of the most powerful tools in the hands of the dwarves. Minecarts also should collect kill lists for a reason if you're not careful.