OK. So you pulverise a million ton asteroid. Then scoop up 0.07% of it in your cutter. Yeah, that makes sense.
No you scoop Tons and tons of stuff to your ship and spill out the waste and keep the Material Wanted thats what refinery does.
So basicly scooping the big asteroid fully is possible leaving behind dust and sand and gravel. Just takes time.
We do Refinery process allready in Earth i dont see the process changing even when we move to space just more efficient methods used (crush raw ore, Pulverize, Reduce, concentrate, filter, smelt/end product)
And blowing it up might be radical but not impossible (example using Nets to contain the fragments, Or using Harmonic frequency vibration to crush it would also cause minimal spread of the fragments, or just cutting the asteroid into pieces with high powered lasers)
Also if we just blow up the asteroid into fast moving pieces we can design the hauler/refinery to be robust enough to withstand the objects.