First, let me commend you for taking the initiative to help develop your potential as a leader. I've worked to help at former employees get promoted on their job within 3 months by walking them through strategic and developmental outlines to move their career ahead. I've trained countless employees in their job duties and shown them ways to improve there skills and complete task more efficiently. After reading your question and concerns, my wheels begin spinning, so I'll share as much as possible in this reply.
First, ensure you have a clear understanding of the difference between a leader and a manager as it pertains to this company. Leaders have the ability to influence another's actions, thoughts, and behaviors. Leaders initiate, give guidance, direction, and empowerment, while a manager is a person responsible for controlling or administering all or part of a company or similar organization. You don't have to hold a managerial position be to a leader, you simply need the ability to influence. Secondly, clearly define your ultimate goal within the organization. Based on the information you provided you should see yourself as an influential employee working toward a managerial position. This will position you to enhance your career within, or even outside the company by focusing on self development while learning the business.
Here are a list of tactics you can follow:
1.) Be dedicated to every task you are given, even if you feel it is outside of your job description. Do more than you are asked to. This will show you can be trusted to get a job done in excellence. Often leaders and managers will give potential prodigies task to see how they will handle them. Don't discredit any task. You never know who's watching.
2.) Develop a leadership work ethic: volunteer for task, be the first to arrive and the last to leave (when time allots based on your work schedule), help other employees when you can, ask to learn new task or information that pertains to the operation of the business or products, offer insight and input when asked or needed. Have an owners mentality and a servant work ethic.
3.) Invest in your personal growth and development. Identify your strengths and weaknesses. Continue to develop and mature in your strengths, and invest in the areas you pin point as your weaknesses. You specifically mentioned having ADHD, speaking "glitches", and low self-esteem. First, don't think of those as disqualifies or barriers. You actually identified your core weaknesses, so you are one step ahead. I would recommend finding a local group, such as Toastmasters (https://www.toastmasters.org), to assist you with focus, public speaking and presentations, and also with boosting your self-esteem.
4.) Continue to develop your relationship and rapport with the owner, manager and other employees. Honor the disposition of those over you by showing you respect them in their roles and show interest in their expertise. Also seek relationships with other leaders or managers outside of your organization and become a student of them.
I hope this begins to point you in the right direction. Generally reaching your goal in this area will take ongoing strategy and coaching, but it is certainly doable. I would love to chat with you and would like to schedule a call to follow up with any additional questions you may have.
I look forward to assisting you in climbing your career ladder,