Had a similar problem today, 11/27/20, with the subordinate eero in the living room, which lead me to re-booting, diagnosing, and updating eero software. It turned out that the problem was in the app (Android version). On the app home page, it showed that eero correctly as Wired, using the Wired icon ("<-->"), but when I selected the location for more detail, it displayed the word "Wireless" and the Wireless icon. For the other eero's it described the connection correctly on both screens: Wired for the main eero in the office, and Wireless for the other subordinate one in the garage. I would have saved myself some time if I had noticed that glitch in the app right off, but I was looking at the detail page and hadn't paid attention to the icon on the home page. It was using Wired all along.

When I pull the Ethernet cable from the subordinate eero, the app changes to showi the wireless symbol on the list page. When I reconnect the Ethernet cable it goes back to showing the wired symbol. So it's not completely arbitrary. But the detail page shows wireless at all time. If you have a laptop with an Ethernet connection, you could plug it into the subordinate eero drectly and run an Internet speed test. If it making a wireless hop, the speed should be slower than when you plug it into your main eero, assuming that  the main eero is on a wired connection to your dsl or cable modem. 

I had this problem where the eero pro was plugged into Ethernet, but showed up in the app as a wireless connection.  The problem is that one wired Eero must be daisy chained to the gateway-wired eero.  Backhauls that lead back to your primary isp supplied router will not work as wired, and are an unsupported topology.

Eero support sent me a list of supported topologies:

https://support.eero.com/hc/en-us/articles/360000830546-Examples-of-common-network-topologies?source=search

My home doesn’t have two Ethernet wires running to any one room, so I’m not able to plug one eero into another in a daisy chain of backhauls, connected by patch cables at the main patch panel.  But after removing the backhaul wire from the most remote wired eero, I restarted all the Eros and waited a while.  Then I measured the bandwidth with a Speedtest and I found the wireless eero was speeding along at wired speeds, above 470Mbps.  It seems there’s little difference in performance when an eero is wired vs wireless.  At least not evidenced in my home with 500 Mb Frontier fiber service.

It’s counter intuitive that the second eero must be plugged into another wired eero.  But on closer inspection I learned that the eeros create a separate subnet with a different IP address range for the wireless network.  It’s a different subnet than the main wired Ethernet coming from my Frontier Arris router.  
 

So the bottom line is, if you want to wire a second eero with an Ethernet backhaul, you must plug that backhaul into another wired eero.