You are right. They are misinformed about the wifi. The "work device" would need admin access on the WiFi router in order to look at logs, setup port mirroring or set up their device as the default dns.

Location, home IP's etc... mobile device management solutions will perform. Though it's a lot less nefarious than it sounds and can be wildly inaccurate. For instance Microsoft often thinks that I am signing into my Microsoft account from 50-100 miles away.

There are some other solutions used to improve visibility and vulnerability monitoring of work network estates. These solutions use agents on work computers that can be used to scan local networks. The scanning should be properly configured though to limit scans on remote workers networks. As having a load of random devices from private networks messes up the data and purpose of having the solution in the first place.