"If Eugene had a public shelter, and a national work program, and enough affordable housing..."

And if a frog had wings he wouldn't bump his bottom so often!

Why should Eugene build a massive public shelter to entice more  'unhoused' people to Eugene?  

Who should pay for it?  And Why?

You want it; you pay for it!

This country hasn't had the 'affordable' housing that you want since the end of WWII and FDR's price and rent controls went away.