Although we're accustomed to seeing many birds alive and well in day to day life, we rarely come across dead birds. Most birds in the wild live, on average, for just a few years, and it's rare for them to die of 'natural' causes alone. Death from old age is generally rare.
Small birds are crucial to the food chain, as they have diets that consist of insects, small amphibians and other invertebrates. They are also predated on by many other mammals and species of birds. This is why they tend to produce many babies, simply because most do not survive to adulthood because of predators. They can also generally breed after their first year - if the winter is survived.
When birds are feeling sick, they will seek out places that are out the way - this behaviour is common with many other creatures. They will often go to the ground and try to hide in a secluded spot. This is because they feel much more vulnerable, and being secluded and having a chance to rest can occasionally help them recover. If they do unfortunately happen to die in that location, the chances are they won't be found in their hideouts.
Scavengers and predators, such as; foxes, rats and cats, will often seek out these hideouts and either take the prey back to feed their young or eat it themselves. This is one of the main reasons we don't find the remains of dead birds. If a predator doesn't manage to find the bird, they rapidly decompose due to their light body mass. This then means that insects will cover the dead body with the remains decaying rapidly before there's a chance to discover it.