Fortigate as a 'nanny'?

Yes, you can adapt the enterprise security features for a home environment, much like you would if you were deploying a FortiGate in a school environment.

At home I have my kids' devices on their own VLAN so I can use more restrictive policies. I use DNS filtering and Web filtering to block things like Adult/Mature Content, Security Risk, and Unrated content (scammers often use pop-up sites to avoid blacklists) and also enforce features like browser Safe Search.  I also block access to botnet URLs (WAN interface setting) and botnet IPs (DNS filter).

However, this isn't going to offer 100% protection against "MS" Tech Support phone scammers who direct victims to legitimate sites and legitimate remote control software.  For protection against this you'd need to go one step further and use Application Control in the Internet Access policy to block most of the 82 pre-defined Remote Access applications.  You can also block things like Tor and Tor2Web here too.

Note that some of the Application Filters require Deep SSL Inspection which involves exporting/importing your FortiGate's CA cert into each device's trusted CA list (covered in several Cookbooks).  This is optional but if you want "maximum" protection you should consider Deep SSL inspection.

Hope this helps,

Russ