(Do I need a sarcasm tag here?)
That still leaves them wearing the same filthy clothes. It doesn't remotely solve the problem. Furthermore, homeless showers tend to be as horrible as most homeless services.
Either you have no privacy -- because why the hell should your dignity matter when you are homeless? -- or it has mold issues, etc.
All services specifically designed to service the homeless population tend to be really, truly terrible in a "you would get sued if this were aimed at middle class people" kind of way. The general attitude is "beggars can't be choosers" and "don't be so uppity -- show a little gratitude for the shitty service you are getting for free, you ungrateful whiner."
So, no, that's not the solution. Missing Middle Housing is The Solution here. Or one part of it.
There is no single cause of homelessness. There is no single solution.