I’m reasonably sure that arguing about leftover toothpaste on the internet does more damage to the environment than leaving trace amounts of toothpaste in its tube.
Argument about toothpaste on a public forum is not about toothpaste. Positive result of such argument is not one person buying less toothpaste but many more becoming aware about wastefulness and how mindless consumption and buying stuff in environmentally harmful packaging and not using it all (and therefore making yourself buy more) causes a lot of harm when everyone does it etc.
Besides, you are using the term "trace amounts" wrong. The amounts people routinely leave are enough for at least a week of toothbrushing.