A lease can legally specify particular move-out requirements, such as carpet-shampooing; since you don't mention a patio power-washing clause, I assume there is no such clause. The obligation of the tenant is (RCW 59.18.130):
(10) Upon termination and vacation, restore the premises to their
initial condition except for reasonable wear and tear or conditions
caused by failure of the landlord to comply with his or her
obligations under this chapter. The tenant shall not be charged for
normal cleaning if he or she has paid a nonrefundable cleaning fee.
Then the question is whether the patio's condition constitutes normal wear and tear, which is undefined by the law. You would be responsible for specific damage that you caused, such as stains and breakage of the surface. If the lease specifically requires you to power-wash the patio on a regular basis, then you should have done that (again, I assume there is no such requirement).
The underlying principle is that you should fix any damage you inflict on the property. If you have dance parties on the roof and damage the shingles, you are responsible. If you simply live under the roof and the shingles degrade from constant rain, you did nothing and are not liable. Likewise if you gouge the paint on the house, you are liable, but if the paint just bubbles up on its own, that is not your responsibility. The question is whether the patio would be in that same condition if you had never used it.
Moss and general dirt are probably within the realm of normal wear and tear; soot on the other hand usually does not develop from natural conditions, though it could, so the question would be, "why is there soot?".
If the court found that a tenant had a duty to maintain some aspect of the residence but the tenant neglected the duty, the tenant could be liable for repairs. But there is no such duty created by statute for tenants, requiring them to wash windows, mow lawns, weed the garden, or anything like that. Since the outside of the house naturally gets dirty, it would be unreasonable to expect a tenant to maintain a rental unit in sale-ready shiny condition. An analog is gutter cleaning. A lease can require tenant to clean gutters, but such conditions are (reportedly: according to Jay Goldstein in this report) rare. So in lieu of an explicit requirement, it is the landlord's duty to clean gutters. The same would hold for patio sweeping.