Seeing that some people already faced this issue & posted about it (here at r/AndroidTV & also over at r/Chromecast), I thought I'd make a post about the solution.
I recently faced this issue too. Right after powering it on, my Chromecast with Google TV started showing "no internet connection" notification, when it was indeed connected to the internet. Going to the WiFi page showed "limited connection" under the WiFi network's name. Speed test showed I have 150+ Mbps speed. Netflix & Prime Video was working fine, but Google apps (& Spotify) wasn't even opening.
Restarting didn't help. Disconnecting & reconnecting to WiFi didn't help.
Then I found out the solution. It was so simple: just manually set the date & time .
Somehow the incorrect date & time was being shown and even though "use network-provided date & time" was enabled, the Chromecast wasn't updating it.
Hope it helps
Thanks for this. Certainly something to try the next time it happens. I was one of the posters complaining about this issue.