The Olympic Medical Center, located primarily in Port Angeles, has not been a "hospital" for many years. As a financial-political maneuver, when it was a hospital, the (then) administrator, Al Remington, determined that the county would enjoy significant financial benefits by demoting the hospital to a community medical center (or some such similar status that does not rise to the status of "hospital"). My understanding is that that is the legal nature of the entity today. At the time of the change, Olympic Memorial did not have a financial interest in what was then the Sequim Medical Clinic. As I recall, but do not know as a fact (although I think that I did at one time), Olympic Memorial acquired some form of interest in the clinic at a later date.
How that may affect the preemption by the state of firearms laws, I don't know. Whether Olympic Memorial actually now owns the clinic, or whether there is some other status of the relationship, is not something I have tracked over the years. If it is still private, then they likely have a lawful right to prohibit the carrying of firearms on the premises. That said, the "add-on" disclaimer to the Brady sign leads one to the logical conclusion that Olympic Memorial has enough of an ownership interest to make the facility subject to the preemption statute.
I believe that the Olympic Medical Center is still governed by a publicly elected Board of Commissioners.