I've traveled successfully on some fairly mangled passports, though never one with the corner missing. Nowadays immigration officials at major airports seem to not even glance at it, they just place it on their chip reader and view the details that way.
Closer examination tends to happen at land or train crossings where such technology isn't available, though I always chuckle thinking back to a Polish immigration officer at Terespol who opened my passport on the page of my Russian visa, thought it was the ID page and flipped back and forth between the cover ('Canada') and that page several times as if he was having an out-of-body experience.