I'm relatively introverted but I took a job managing a team of about 15 which I grew to 30-something. Many years ago before I was a manager I was talking with a friend (a headteacher) about it and said I was worried about the responsibility that came with management, having to deal with problems etc. He made a very valid point which was that yes you have more problems to deal with, but also you are empowered to deal with them - so in some ways it's easier because you have more influence. Unfortunately there will be some things you can't influence sufficiently but that's just part of management, being able to accept 'bad decisions' made outside your control, explain them to the team (even if you don't fully agree) and move on.
As mentioned above sometimes people tend to think of management positions purely in terms of people skills but leadership is also about:
- Articulating and disseminating the vision
- Being clear about what is expected of people
- 'Fighting the corner' for the team i.e. listening to concerns and working to address them
- Being a sounding board
- Sound decision making
I also found one of the more rewarding elements of management that I hadn't really anticipated beforehand was seeing people progressing, e.g. hiring someone in a junior role and then getting them a promotion. To the point above about influencing things, as the team grew I was able to implement a hierarchy under me for the perms so they had clearer development pathways whilst also giving some of my team management responsibility (and keeping my number of direct reports in check). So I actually came to enjoy the people development aspect of management and I contributed a lot in leadership team discussions around performance appraisals, employee engagement etc etc.
I think it's a misconception that managers should always be extroverts; charisma can help with things like instilling confidence, setting out the vision etc but conversely introverts can often have strong decision quality, thinking through the consequences of something rather than charging in to a rash decision.
Finally you will find that as with most things, the more you do something the more confident you get with it. When I first started interviewing people, I was a bit nervous myself and did quite a lot of prep work, but over time it started to come quite naturally (stuff like explaining who I am, what the team does, the company etc just rolled off the tongue). The same with running big team meetings, if you are used to sitting quietly in the corner chipping in with the odd comment it may feel a bit daunting but get a few under your belt and it will feel more natural.