Washing the egg removes the poo and the coating.
No source provided and this may just be some myth.
This includes salmonella, which may be present if your flock is infected in the poop on the outside of the shell (remember hens only have one egress port), plus any other sources of environmental pathogens, of which there are many.
When the bloom is washed off the egg, pathogens have an easier time penetrating the shell and consuming the nutritious yummy bits inside. At room temperature, they can multiply rapidly. Refrigeration slows the rate of growth.
An unwashed egg retains the barrier, and stays fresh longer without refrigeration.
YMMV on household acceptance of dirty eggs on countertops, but they are cleaner than many other items within arms' reach that we are conditioned to not think about. :)
Incidentally, I heard somewhere that using a ridge to crack eggs on (like the edge of a frying pan) isn't best as that can possibly drive a bit of poopy shell into the interior though if it's just about to be cooked and eaten then that's less problematic. I use the flat kitchen top to crack the shell instead which leads to the occasional amusing outcome of cracking it too hard and dumping the whole egg onto the worktop.
They will easily last 4-6 weeks with no major degradation (i.e. still good for omelets, but use the freshest ones for poaching).
The forest predators eventually help moderate our egg surplus. Free range comes with risks, alas.
Interesting point about ridge-cracking. I'd never thought about it, but it makes sense and I will mend my ways! :)