...I never got the sense this was meant to be good thing. Sure, he may leverage it in some incarnations but most time, it's clear that he'd rather travel without needing to be recognized. Learning he influenced a culture so "Doctor" meant "Warrior" was likely not what he liked to hear.

Plus... his travels wouldn't go unnoted as planets and space goers hear about this blue box carrying a man with a brain that can save the world. Even without his infamy during the Time War.