Just a couple additional thoughts on this. First, if you've already set up the new to-be-remote unit near the primary as a wireless unit and verified it's working that way, then I would connect it via Ethernet to the primary unit's LAN Ethernet port (still nearby) and verify in the Google Home app that the new unit shows as having a connection type of "wired". Next, I would use a pair of inexpensive, unmanaged Ethernet switches, one in the main house connected to the primary's LAN Ethernet port and the other in the outbuilding with the long cable run between them. This provides some electrical isolation to protect the more expensive Google devices in the event of a lightning strike. It also provides more Ethernet ports in the main house that can be used to build out a wired network, and in the outbuilding to connect wired devices out there. Once those are in place, you can test that network in the outbuilding with a laptop or something to verify the wired network is working reliably. Then connect the new remote Google WiFi unit to the Ethernet switch in the outbuilding and verify it joins the network and shows as having a wired connection.
If this doesn't work reliably, the most likely issue is the Ethernet connection, which is why I recommend testing those components independently, and one piece at a time so you know where the issue is (rather than skipping to the end and wondering why it doesn't work).
Avoid smart/managed Ethernet switches when you build a network like this. They will cause problems due to the loop detection protocol Google/Nest WiFi uses.
