The problem I have with garbage dumping is that stone is heavy and takes lots of Dwarf hours to move, even if the dump zone is close. The wheelbarrow method is flawed because the Dwarfs end up hauling all the stone instead of just the rocks you need them to haul, and the fact that you kind of need space (which you're trying to get) in the first place.
You didn't mention the size of the rooms you wanted to clear the stone from, so they must be fairly large, and it sounds that you are not interested in setting up a minecart QS for stone, so consider this...
You get about 1 stone for every 4 tiles dug.
So if you want to use wheelbarrows and you do not want to use a minecart quantum stockpile, then you need to use 20% of the area you dig out to both store the stone and clear the room. If the room you want to clear is 60x60, then you need about 30x30 for your stone stockpiles.Stone stockpiles take the most recently dug stone first.
So if you have dug out the stone storage room first, and put 6 stockpiles in it for stone, each with 3 wheelbarrows, you can have 18 wheelbarrows hauling stone at the same time you are digging out the 60x60 room.If I was setting this up, with the mason's shop near the surface and the smelters near the magma, I would split the 6 stone stockpiles into a group of 4, just below the mason's shop, and a group of 2, just above the smelters, and set them to accept the appropriate stone types.
Then you will have something like this:
100z - surface/masons (the stockpiles adjacent to the workshops would only take from links and only take from the stockpiles on 95z)
95z - the four common stone stockpiles (these will take from anywhere and give to the stockpiles on 100z)
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50z - the central stockpile room
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0z - the two stockpiles here would accept metal ore and flux (these will take from anywhere and give to the stockpiles on -10z)
-10z - magma and smelters and forges (the stockpiles adjacent to the smelters would only accept from links and only take from the stockpiles on 0z)
The primary concept is, "get the rock sorted immediately and get it moved while it is being dug". You can have rooms cleared of stone quickly by having more wheelbarrows available for use than miners who are mining.