There's a distinction between what I think of as "leadership" and the basic requirements of functional adulthood you describe here.

Leadership, for me, is associated with:

- strong future orientation: obsessed with a better imagined future at a macro/holistic level e.g. corporate/industry rather than team/dept level

- the discipline to invest time/energy into honing skills even beyond the current role

- excellent communication skills; a diplomat who inspires others

- action orientation: does it, rather than just talking about it.

Anyone, at any level, can develop these attitudes and skills: there are plenty of online courses where you can learn, even for free, the leadership skills taught in business schools. You can initiate/volunteer for groups at work such as the charities committee and hone leadership skills there, or just day to day within your team on projects.

You are, of course, right, about self-management. A core leadership skill is developing a certain composure: some of us are naturally very expressive, and that's fine - we are not automatons - but taking a deep breath, thinking before reacting, and soliciting input before sending a reply are all important habits.