Seattlites are desperate to conflate the housing "crisis" with homelessness, but I'm not convinced the person yelling at the sky and swinging a drain pipe at people in his alley narrowly missed his $5000 mortgage payment. It's unfortunate that people with less means have to live further outside the city and spend more of their commuting, but I don't it's unreasonable to assume that people will make decisions on their own behalf to stay housed rather than just live on the streets if rent is becoming too expensive.
It's plenty cheap in Renton. I think it's unreasonable to put a homeless shelter in the middle of downtown Seattle knowing that they're working with a limited budget and that paying a premium on the lot will result in less beds.
There are two issues with this, and they both go to the heart of why developing these services is so difficult:
1) Seattle residents have chosen to tax ourselves to pay for social services. By and large, a taxing jurisdiction can't put services outside its jurisdiction, so Seattle can't levy a tax and then use it to buy space in Renton. Yet, Seattle is more or less the only jurisdiction in King County willing to tax ourselves to pay for these services. The Eastside for damn sure isn't, at least not for services to people who aren't sympathetic groups, and would rather simply arrest people for vagrancy and haul them off to the King County jail (which is, per state law, in the county seat, thus Seattle) and then conveniently be somewhere else when the person bonds out of jail and walks onto Seattle's streets.
Arguably the South King cities don't have the tax base to help shoulder the load, Renton being one of the few that does, but that then gets into people not being willing to vote for those taxes because, frankly, they don't want to. So you have a situation where only Seattle is spending the lion's share of the money but Seattle can't spend it outside of Seattle, land is expensive virtually everywhere in Seattle, and King County's government has shown precious little impulse to try to handle the problem at the county level. To say nothing of Washington State's almost-total abdication of responsibility.
And then,
2) People get incredibly vocal about homeless shelters and other services for them. Just try putting a new shelter in Wallingford. Hell, just try putting one in Bellevue's Eastgate, like they've been trying to do for a decade. The lawsuits alone are just now starting to wrap up and the vocal minority, realizing they've run out of court-based options, are running candidates who sit on a platform of "yes it is needed but not here" and have no answer for "but where?" because that's not the point.
I'm sure the good people of Renton would be vocally opposed to your idea to "export Seattle's homeless problem" to Renton. Thus, the only areas that wind up with shelters and other public services are the ones who either lack political clout--like Pioneer Square and the ID--or are where those services have always been...like Pioneer Square and the ID.
Bellevue actually kicks money back to Seattle for homeless shelters, along with running their own (but near east gate, not a great locale).
Yes, Bellevue cops aren’t as soft as Seattle cops (they send riot squad out if some guy is sleeping on a park bench), but it is still a liberal area.
Renton on the other hand...is basically a giant strip mall, homeless people wouldn’t find much their even if they did setup shelters. And they already have a lot of questionable characters in what passes for their downtown area (I’ve taken the 565 a few times between Bellevue and the airport through Renton).
I agree, spending tax money collected from one area to spend in another is a problem, but that's not to say that there's no place in Seattle proper that isn't cheap enough.
To the second point, there will always be NIMBYism, and to be honest the expensive parts of Seattle will fight it a lot less than the cheap places, but I don't think that directly rebuts the point.
