Thanks for your replies, @koying !
The problems appears to have been with my wired network and routers not forwarding between VLANs. I'm not quite certain what isn't forwarding, since things like Google Home forward correctly, but solving that problem is for another day. 
Once I ensured that everything was connected to the same VLAN, it all seems to be working.
I must admit that I'm a little disappointed with the new Thread Network Mesh display. The lack of a on-screen legend makes it difficult to decode, and I had to go to their GitHub to find the meaning of the symbols. Simple flyover text popups would be very useful there.
The mesh display, through the dashed line, is trying to say that Test Product is connected to the Home Assistant device through a "link to an offline or unknown/external device". Looking at the nRF Topology Monitor screenshot, that does make sense -- the Test Product is communicating via either the Nest Hub or the extra Pi OTBR, and the mesh display does not know either device.
What I find odd is that Settings -> Thread lists all three border routers. Home Assistant is clearly aware of the extra Pi OTBR as well as the Google Nest Hub that are part of the network, but that information is not making it into the Thread Network Mesh display.
Perhaps I'm missing something about passing that border router information to the Thread Network Mesh display, but the fact that the mesh display is missing the extra border routers seems to defeat the purpose of helping people understand their Thread network connections. You'll see the quality of the connection from the color of the connecting line, but without knowing what device it is connecting to, one would not know how to move a device for better reception.
That all seems like it should be in a new posting, though, since it is not on topic for this post.