Short answer: No
Decomposition is an aerobic process, it needs oxygen to happen - that's why we continually turn our composts piles. Well preserved fossils were formed in an anaerobic environment - without oxygen, instead of decomposing, they turned into rocks. You won't really have fossils in your garden, you will have a landfill. Although archeologists do get pretty excited when they find an old dump.
If you choose to leave your kitchen waste on the surface of the soil, it will decompose but, the bacteria will be using nitrogen in the process. So, the soil will be low in nitrogen until the decomposing process is completely finished. When the bacteria finish the job and die, the nutrients will be put back into the soil. In the meantime, you will have a messy, rat filled garden.
In the desert, you have to water compost piles almost daily or they sit for years without decomposing. I have never been a fan of the closed bins as they don't allow enough air circulation.