Earlier quoted context omitted.
That default, sadly, would basically guarantee the thing doesn't work for all too many users. And as a consumer electronics product (especially in the sub-$50 price-range), the market-smart thing to do is configure the defaults to work in the saddle-point of worst-case and common scenario (i.e. badly-configured local router talking to a standards-hostile ISP's DHCP configurations).
The proper thing to do is to use the DNS settings the DHCP server provided and testing those settings by providing a server the device can lookup and connect to (with TLS). If the server proved it's authenticity, the DNS settings work. (some devices might cache this result, others might do this during startup) If an error occurs or a reasonably short timeout expires, the device can: if it has UI the user will see, it…
How? And what should the user do with that information?
This device is not architected for users who know what DNS, DHCP, or TLS are, much less who care.