No matter how you define it, a public school is always a government school. Some may be less indoctrinating than others, but in the Seattle area I can’t imagine any being non Marxist.
Once outside of King County, things are somewhat “moderate” with the exception of Thurston County (Olympia).
South of Olympia, pure redneck.
East of the Cascades, pure redneck.
North of Everett, mostly redneck.