I've been thinking about this a bit. It seems to me that, for wheelbarrows and mine carts to be a significant component of the actual excavation process, it needs to be necessary, or at least strongly desirable, to clear out a space before mining the next tile. Otherwise people will just ignore the mined-out stone until they have a use for it, whatever form it's in, like they do now. This could be done by making mining require an empty tile where the miner stands, or by making tiles cluttered with rock drastically slow down mining or movement.
I also think it would be useful to differentiate between usable chunks of stone and gravel/rubble/mullock/whatever. As Kohaku points out, we need a way to deal with waste stone that won't murder our FPS, either efficiently removing it from the map or converting it to a form that doesn't contribute so much to the buildup of items that must be tracked. My though, for what merit it has: mining could create a unit of rubble at the mined-out tile, and a unit of rubble or usable stone at the miner's location, depending on skill. Two units of building material would be needed to construct a wall, while one would be enough for a ramp, floor, staircase, etc. A unit of rubble in an empty tile would form a natural ramp of (type of stone) (rubble/gravel/etc.). Adding a second unit of the same type of rubble would convert it into a natural wall of that sort, while adding a different type would convert it into a generic rubble wall, with no material information saved. Optimally, dumping rubble onto a ramp next to a cliff would cause it to fall off the side of the cliff, forming a pile below, but I suppose that stable pillars of rubble make as much sense as walls made of soap.
Rubble of a metal ore could still be smelted, so players wouldn't lose metal if a low-skilled miner dug out part of an ore vein unless the rubble got mixed in with waste stone. Rubble could be used as fill between solid walls, dropped into rivers and lakes to build dams, causeways and the like, or just piled up in a valley somewhere to get it out of the way. As aka010101
noted, it could also be used to make gravel roadways, which would eventually wear out and need to be replenished. Abstracted piles of tailings off map would probably be reasonable.
I think it makes sense to have alternate designations for quarrying, in an effort to get as maximize the usable stone produced, and excavating, to turn solid rock into movable rubble as quickly as possible.
Naturally, any attempt to make dealing with waste stone an important aspect of excavation will make it slower and more difficult to dig out a fortress. Some players will consider this a good thing, and others a bad one. I don't know what would make both sides happy, besides maybe an init option to turn rubble off.