What do developed nations do with organic municipal waste? Are there separate receptacles in addition to bins for plastics, glass, metals, etc.? Is organic waste then transported for anaerobic digestion or composting? By "developed nations" I mean "nations that have developed and sustainable waste management systems" (not the two-bin US or the incinerating Japan, for instance)
I don't know that any nation has a uniform system of domestic waste collection and processing/disposal. Waste collection and disposal systems are usually determined by municipal councils, either individually or in collaboration with neighboring councils.
For the past 25 to 30 years, depending on where I have lived, my experience has been each household is supplied with three colored bins by the local council for curbside collection. One smaller bin is for general household rubbish that will be either dumped in to landfill, burned in power generating incinerator or "processed" by other means. This bin is put out each week.
The other two bins are for recyclables and for garden waste respectively. These bins are put out on alternate weeks. On odd numbered weeks it might be the recyclable bin and on even numbered weeks it would be the garden waste bin that gets put out. In addition to this, some councils have a system where they collect larger amounts of garden waste twice a year. The councils inform householders at the start of the year, on which two dates, six months apart, it will collect additional garden waste for their region. These dates are usually in spring and autumn. For a week prior to those dates, householders can prune, chop and tidy their gardens and place the additional garden waste in large heaps on the curb or verge outside their houses.
Upon collect, all garden waste, is turned into compost or mulch which the council then sells to whoever wants to buy it.
Generally the best uses of organic municipal waste is by extracting whatever biogas you can, created in anaerobic conditions, and injecting that biogas (methane) into the natural gas network. Generally you can't make enough methane that way so only small percentage of the total used methane is biogas and most of it is still natural gas, but it's a very useful way to reduce the use of the natural gas somewhat. Also the organic waste is then decomposed and becomes new soil used for gardens for example.
That's for sorted organic waste. For non-sorted waste, it is directly burned to energy, and that energy is used to create district heating (if in a cold area where heating is required) and electricity. If in a warm area where heating is not required, only electricity is produced.