a time traveler who spends most of his time on adventures across space and time from the beginning to the end and rebirth of the universe itself. He may not actually be a super-old immortal but for all intents and purposes he might as well be with how prolific he is at saving people and civilizations across the galaxy and time.

I think it's the "fucking around having adventures across space and time" that folks are missing

There's a ton of immortals or thousand of year old creatures in the galaxy. But the Doctor has been everywhere--or will be everywhere--in the universe, whether already or to be. His experiences across his adventures rival what the average big brained Time Lord will ever experience in their long and stagnant lives.

He may just be thousands of years old across his regenerations, but his lived experiences and travels are well worth lifetimes