On my wired Google WiFi system, the status endpoint also shows ethernetLink as false in the "wan" section, but shows lan0link as true in the "system" section. In any case, if the Google Home app shows the connection type as "wired" then you have them connected via Ethernet. You have to connect the switch through the LAN Ethernet port on the primary Google WiFi (router) unit. If things aren't working well, then I would investigate the wiring and the type of Ethernet switch. Managed switches often have "loop detection" (spanning tree protocol) enabled in a way that interferes significantly with how Google WiFi deals with the loop introduced by wiring secondaries. I usually recommend using inexpensive (unmanaged) switches, but you may also be able to disable STP on your switch. The other potential issue is just that the wiring may only be managing to hold 100Mbps rather than 1Gbps connections. If you still have the old Google WiFi app, you can run a mesh test there, and while still on the results screen, open the "..." menu and select "Details" to see the speed each secondary is managing to talk to the primary. For wired secondaries through 1Gbps cables and switches, these should a few hundred megabits per second (it's a simultaneous stress test, so these numbers can only be compared to other mesh test results – they aren't a "speed limit" for normal operation). If the numbers in here are 100Mbps, then test the wiring all the way from each secondary, through the switch, and to the primary's LAN port. But, if the numbers are only a few megabits per second, that would make me suspect the managed switch issue is more likely (although it could also be a very bad Ethernet cable).