As I recall, Mycroft was described as brilliant but lazy.
And they never talked about their ‘implied’ older brother.
In the traditions of the time, the oldest brother inherited the title (there is some hint of the father being a minor Lord of the Realm). If so, then the second son, Mycroft, would be expected to go into Government Service, and, the third, Sherlock,would be cast out on his own to be a wastrel or find a job. In “A Study in Scarlett” he had to find a roommate to help pay rent, and, in “the Priory School” he receives a hefty check from the Duke and says he is a poor man. Of course, after that hefty check, he was no longer.
I think “lazy” is unfair..
“Uninterested” or “couldn’t be bothered” is more accurate.
He worked on things he cared about quite hard.
He was just a mythanthrope (sp?), which I fully appreciate.