Thread is an IPv6 bearing mesh network; and if you have a border router that has the appropriate IP routing rules, and your firewall allows it, your Nanoleaf bulbs can most certainly reach out to the internet (granted, they only have the concept of IPv6, so a majority of the internet is inaccessible if only because it is IPv4)

Any ideas if the HomePod Mini (probably the most popular Thread border router) grant broader internet access to Thread devices on the mesh network? I can't find any clear info about how this might work or how to test.

"If you allow the devices to access the internet they can access the internet" isn't really as doom and gloom as the parent is suggesting.