Should a local (online tutoring, no physical business address) site even consider Facebook or Godaddy as their main site?  Even with a good domain, most of the new clients are coming from direct referrals and probably social.  I'm thinking Facebook might do as well as a brochure web site.

As the main site, no.

GD, No way.

A fb page would probably work, but might lose some business because of not having an actual site.

She has good domains, I was thinking she could redirect to an FB page.

Her host just moved and whatever rank she had is gone.  Besides, tutoring is like locksmiths and serp reflects that.

Let's back up. Why do you ask?

- simplicity? No tech to watch over?
- cost? Free?

Because, for example, she would almost certainly be within the free tier for Google App Engine and have it resolve to her own domain. But you would need to find a tech head to set it all up.
https://cloud.google.com/appengine/docs/standard/python/mapping-custom-domains

If you're not picky about the look and feel, you could use your own domain for free with Google sites
https://www.siteground.com/tutorials/googleapps/google_sites.htm

>- simplicity? No tech to watch over?

Mostly that.

A brochure is good enough for her business. She's picked up a few clients from the site, though I can't figure how they managed to find it. 

Without steady immersion in the tech/skills needed to manage a site, the owner forgets everything about cpanel, etc. and the site dies.