Hello @cynops
The first key is to make sure your wired secondaries are connecting in to a wired network rooted in the LAN Ethernet port on the primary. This is the "inner" network created by the primary, which is a router/firewall. You can't connect them all to the "outer" network (the one the primary's WAN Ethernet port is connected to).
Secondly, make sure any Ethernet switches you use on that inner network is an inexpensive (unmanaged) switch. You may be able to make a managed switch work, but you will need to do some reconfiguration of it to work around a loop detection (spanning tree protocol) conflict.
I would also put inexpensive switches on the ends of any long cable runs, especially any that go between buildings. That's mostly for safety reasons (to protect the more expensive Nest WiFi Router units from any lightning strikes or other power surges), but it can also help with debugging, since you can connect a laptop or something to one of the other leaf-node switch ports to make sure the wired network is working properly.
Lastly, to tell whether your wired secondaries are using the wired backhaul properly or not, launch the Google Home app, tap the "Wi-Fi" bubble, then "Wifi devices (#)", and then for each secondary under "Nest Wifi points", tap it and hit the gear icon in the upper right corner. In the screen that opens, in the section titled "Wi-Fi information", the "Connection type" should be "Wired". If it isn't then you have something not quite right in your Ethernet network. It should "just work" if you get it connected properly.
If you need to debug, I would build things up one piece at a time, and check the Connection Type in between each step. For example, connect one secondary directly to the primary's LAN port with a short Ethernet cable. Then add your core switch. Then check the long cable with a laptop. Then connect the secondary to that known-good long cable, etc. That way, you'll know where any issues are and can focus on fixing that.
Good luck!
