> My biggest issue with Thread is (AFAICT) it gives each device access to the internet.

Thread is a protocol, this is only as true as the statement "WiFi gives each device access to the internet".

It _can_ give devices access to the internet, but only if you plug your (border) router into an internet connection. (Or otherwise bridge the networks. Your border router could still have internet access itself without bridging the thread network out to the internet.)