What do you mean (or what do you think Merriam-Webster means) by cleanliness?
Typically, when people say to their kids, "clean your room," they do not mean scrub the walls, sanitize the surfaces and they may not even mean vacuum the floor. What they almost certainly mean is remove any visible mess: make the bed, put away the toys, put the dirty laundry in the hamper, etc.
If you say to your kids, "clean your plate," you either mean wash it or finish eating the food that's on it (in which case, it may still be very dirty).
If a shave is "clean," there are no untidy hairs, but this says nothing about any germs that were on the razor and have now been spread to the face.
This is something of a circular reference. Neat and tidy might mean clean because clean means neat and tidy.
The meanings of cleanliness are more diverse and complicated than it may seem. It will probably not answer your English language question, but Mary Douglas's Purity and Danger is definitely worth reading.