Phone suddenly turns off even if the battery is about 80% charged

Have you tried replacing the battery? The Nexus 6 was announced in 2014 which is now well past the expected usable battery life. When the battery is very old, it will not be able to provide the extra power that the phone requires for the radio, CPU and screen. The result is crashes and power offs.

You can confirm this by fully charging the phone, putting it into airplane mode and observing how long it stays on before reaching 80% and if it shuts down at that point.