The first Nest WiFi Router set up as part of a new Nest WiFi system must have an Ethernet connection to your internet service. It is not a WiFi extender. It is a router+firewall that also creates a wireless mesh to which other Nest WiFi nodes (more Nest WiFi Router units, Nest WiFi Point units, or Google WiFi units) can be added.
I just purchased an additional wifi nest device Google WiFi System, 1-Pack - Router Replacement for Whole Home Coverage - NLS-1304-25 (Renewed). I'm trying to set it up in a different room, but directions in the Google Home app tell me to plug in ethernet. I did not have to do this before with 3 wifi points not using ethernet. Did I purchase the wrong type of device to add a wifi point?
It sounds like you haven't migrated your existing system into the Google Home app. That will be a necessary first step, since the Google WiFi app can't be used to make changes now: https://support.google.com/googlenest/answer/9547597?hl=en
Once you've done that, you should be able to follow the "Add an additional point" instructions here: https://support.google.com/googlenest/answer/7183148?hl=en
I already have the system in Google Home app. In the Home app, I could see Google Router, and the 2 points I already have in living room and office. I want to set up an additional point in a sunroom - that is what I was having trouble setting up. In the 2nd link you shared https://support.google.com/googlenest/answer/7183148?hl=en when I got to the point of scanning the QR code, multiple tries, and the next step said to connect ethernet. After 2 more attempts just a few minutes ago which asked me to connect ethernet. Now on my 3rd attempt this morning, the set up did not require that ethernet part. But 3rd (or 8th) time is a charm - for whatever reason on this try the set up adding the new device as a point worked for me ¯\_(ツ)_/¯. Thanks for the info MichaelP.