I seem to have resolved this issue with my set-up. Fire HD 10 (9th Gen) and my ISP is Virgin Media so I have a Hub 3.0 providing WiFi. This fix may not work with other providers.
Having followed all the checks above, several times, and reset the router a couple of times for good measure I was still left in the same position - tablet connects to WiFi but with message “Connected, no internet” despite other devices connecting without any issues.
A few years ago there were big security issues with a protocol called Universal Plug and Play (UPnP) and recommendations from various internet security sources to turn it off. I was not expecting anything spectacular but as soon as I disabled UPnP on my router my Fire tablet started pinging messages at me regarding updates and messages from Facebook!