Is there any software for checking hardware in iOS? mentions a few things for iPhone, some of which may work on iPad. But I can't get them, because the iPad has no internet.
With WiFi enabled, it connects to the AP but does not display the WiFi icon and TCP apparently does not work.
When I turn off WiFi, I still have LTE, but TCP still does not seem to work. Safari, Stack Exchange, SpeedTest, etc. However, in a terminal, ping 188.8.131.52 sent and received 18 packets with an average round trip of 20 milliseconds.
Apparently—since I can ping IP addresses and (update!) I can surf with Tor—TCP is working but DNS is not. Yet one of the many things that has already been tried is "Reset Network Settings."
UPDATE: It was not only DNS, but it is still a mystery. Both problems survived changing the WiFi DNS address to a known good one and another that was probably good, rebooting the iPad, rebooting the WiFi router, and resetting network settings on the iPad. But the WiFi problem and the cellular outage were both fixed by "forgetting" the WiFi and re-joining.
But the question is still whether there is any software that can be run on a Mac to diagnose an IOS device.