We have those; they're called landfills.

> neatly separated pile[s]

The problem is that most types of refuse are already semi-uniformly mixed at the household level, so you would have spend labor separating them, which might not be needed[0], so you might as well put off separation until know you need it.

0: Eg, if the most efficient way of recycling them turns out to be: dump the whole mess into vat of genetically engineered bacteria, and sift out the glass and metal bits when they build up.