It's not self-bullshitting. In many smaller companies the programmers already have to play the roles of the BAs/PMs.
Can you say the opposite for BAs/PMs? These positions are introduced after an organization or project reaches a certain size. It's simple specialization, you don't want everybody to do everything. When this transition happens, some programmers might become BAs or PMs. Alas, what about the opposite? How people that started as BAs ever make the transition to programmer? There is an obvious trajectory here.
You're talking about tech companies where the business is basically programming. In a company that builds cars or medical devices or a bank, a programmer is about as likely to become a BA as the other way round.
As for project management, that's, well, a management position. Tends to have a better salary, prestige and career prospects than non-management positions. That fully explains why many people want to move into management and hardly anyone who's chosen it as their original career wants to move out of it.
Because in those companies, a BA needs in-depth knowledge of the domain that is as hard and slow for a programmer to acquire as it is for a non-programmer to learn programming.
Can you say the opposite for BAs/PMs? These positions are introduced after an organization or project reaches a certain size. It's simple specialization, you don't want everybody to do everything. When this transition happens, some programmers might become BAs or PMs. Alas, what about the opposite? How people that started as BAs ever make the transition to programmer? There is an obvious trajectory here.