Google Nest Community

I can't connect to the wifi through ethernet. I can connect through wifi, but when trying through ethernet it just says no internet on the ethernet section and connected on the wifi section. I have the cord connected on the grey <> side, not the blue...

I have a first gen mesh wifi setup with 2 access points, and I want to get a third. The nest points/router have much better speeds. Would i get those speed by adding a nest access point or router to the current system or would the speeds be throttled...