Let’s put it this way- “City of Seattle” has the same meaning as you would say “Seattle City” in any instance and whatever level of correctness it contains.
I’m gonna go with a no on that, at least in terms of what to call the place name that is Seattle. In Oregon, there’s a city is called Oregon City. If you were to call it “the city of Oregon” people wouldn’t know what you were talking about, at least not with any amount of immediacy. How does “the city of Seattle City” sound? If you say Seattle with any regularity (say you live in the NW of the United States or you work for Boeing), it will likely sound pretty odd given that no one calls Seattle “Seattle City”.
Additionally, while you can take “City” off of “New York City” you can’t do that with “Oregon City”. If you were to say “I live in Oregon” the assumption is that you’re talking about the state as a whole, not the a specific city. In my experience (as someone who is from Oregon and tells people “I’m from Oregon” on a someone basis), no one is going to ask for clarification in terms of whether or not you mean the state of Oregon City. Meanwhile, there is some ambiguity when you say New York.