Should a Talmid Chacham even avoid using a public toilet, so people do not see him going in and out and fell he is too relatable?

Listen, this is a misunderstanding of what it means to be a Talmid Chacham. The issue is not about avoiding public toilets to seem "less relatable" - that's nonsense. A Talmid Chacham is a human being who has to go to the bathroom just like everyone else. In fact, one of the things that's supposed to eliminate all pride and arrogance is the very fact that you have to go to the bathroom. What are you so proud about? Your entire body consumes and ejects waste. This reality should remove all ga'ava from a person. The real halacha is different. A Talmid Chacham should not walk into a non-kosher restaurant to use the bathroom. Why? Because when the public sees you walking into McDonald's, they're not going to think "Oh, he's going to the bathroom." They're thinking either the food there is kosher or the rabbi is not kosher. They're never going to calculate that you're just using the facilities. This creates a Chillul Hashem - a desecration of Hashem's name. But using a regular public bathroom? There's no problem with that. The concern is about how your actions appear to the public and whether they could cause people to stumble or think wrongly about Torah observance. A Talmid Chacham has to be careful with his actions in public to avoid any Chillul Hashem, but this is about preventing misunderstanding, not about maintaining some artificial distance or appearing superhuman.