That's lovely sounding, but false for a couple of reasons.
1) There's a capability gap. Charities can't do a number of things that are needed that local government can. Simple things like permission to operate portable toilets, or working alongside cops for redirection away from jail it possible, are somewhere between multi-month activities outside their core-competencies to impossible for private actors.
2) The vast bulk of private charities segment their 'market' in various ways that leaves gaps. That's fine, but it isn't going to "mostly solve" the issues caused by society-wide economic failures.
3) Because the problems are so varied, coordination reaps substantial benefits. Social workers talking to cleanup crews talking to mental health talking to cops is a dynamic that your local churches are not equipped to take part in.
Kneejerk antigovernment bias is a national pastime, I know. I'll even play along when it is valid. But the idea that the Salvation Army is going to spring to the rescue is fantasy.