It has also been tried many times to separate organic matter from the rest (different colour of bags), and using automatic systems to sort it. It has been mostly dropped because it's more expensive than just burning everything unfortunately. Which brings me to the main point. We can be very efficient and recycle most of the waste, find a optimum point where we minimize the environmental footprint - it's just expensive, and the first thing to go when the city need to save money. People just don't care about future generations, or don't believe it's a problem, to accept the cost of recycling.
But it really is a shame, because high density areas like Copenhagen, are ideal for pushing recycling. You get quantities, and you don't have to collect over a large geographical area. So the running cost of collecting is significantly less, and you can build larger and more efficient facilities to process and recycle the waste.
For a long time we used to have general waste and compostables. That was quite understandable because you really want the compostables into a separate heap because of...well, composting and you can produce bio gas off composts if the percentage of decomposing waste is high enough. But I'm pretty sure the separation of glass, wood, papers, cardboards etc. could be done in some automated way that reaches a good enough success rate that would make it economically and ecologically viable.