The page you linked to didn’t say they “wouldn’t be admitting more Asians than the population they are picking candidates from” nor did it even remotely imply that.
The easiest way to get into UW is to be (graduate high school) from eastern Washington, they actually have a state mandated quota for that.
If they have that aspiration they have tools to implement it. The tool is called "holistic review", which opaquifies the admissions process.
People who read college applications are "guided" by Universities using a system of feedback so they can get whatever outcome they want. Read more about that here:
UW is still a state school controller indirectly and sometimes directly by the state government. The reason the Easter WA quota exists is a because of its charter. The administrators at UW can express whatever opinion they want but don’t rule by fiat, it would take probably a state legislative vote to implement the quota you imagine exists.
College admission officers don’t get access to race and such when reviewing applications. Yes, they can add points to it, but that happens after the rest of the application is reviewed, and can’t be done without explicitly saying they are doing it.
Because of "holistic review", admissions are subjective, which means admission officers can do whatever they want. Admission officers do have access to race, why else is it on applications?
Read the NYT story I linked to, to understand how application readers are gently guided to consider race, without explicitly being told to do so.
Excerpt from the story:
"I received an e-mail from the assistant director suggesting I was not with the program: “You’ve got 15 outlier, which is quite a lot."
The easiest way to get into UW is to be (graduate high school) from eastern Washington, they actually have a state mandated quota for that.