Ok, some facts.
On average 8.8 percent of the material that is collected for recycling ends up in the landfill. Almost all of that is material that should not have been put in the bin in the first place like plastic tubs.
It is not a good idea to give everyone a back yard compost bin. Home composting requires some degree of commitment and attention. It is not realistic to expect the majority of folks to take the time needed to compost on their own.
Also, the organics collection offeres the option of composting lots more material than you can do in your back yard including contaminated paper such as napkins and paper towels, meat, bones, sauces and gravy, pet waste, and disposable diapers.
The collected material would be composted in a digester that would generate methane that would be captured and burned for electricty or converted into truck fuel. Can't do that in your back yard.
Kid Capital wrote "It would get disgusting and reek really really bad. " It would not stink anymore than your regular garbage bin does now. Its the same stuff, just in a different cart.
The collection of the organics would be done every week. The rest of the stuff that is not composted or recycled would be collected every other week, perhaps monthly. We would not need to add a lot of trucks to get that collected.
Some of the carbon generated by these extra trucks can be offset by picking up grass clippings in the summer cutting down on drives to the drop off sites for the folks who still bag their grass.