You should sell those experiences as first steps into being a leader.
However, you should be aware that managing and coding are so different, even if it's in the same field. Managing people is tough and you learn it by doing.
It's the typical hiring chicken-and-egg problem: You need experience doing X to get a job doing X. People management is no different.
In addition to taking on more leadership roles in engineering projects, I began finding and taking on opportunities on the side that have a managerial element to them, for instance recruiting (sourcing, behavioral interviews, selling candidates, redesigning our interviewing processes), mentoring/coaching people, driving our company's internship program, etc. I found that I enjoyed that sort of nontechnical work. In particular, I liked thinking about how to grow other people and help them be effective.
I vaguely started thinking that I wanted to just switch into management, and I advertised that fact (mostly to managers). I wasn't too urgent about it, and an opportunity showed up [after a few months] and the management team thought of me first as someone who could be a good fit. Part of what made it a good fit was that the team was not 'on fire' or anything, so if I were to do a bad job or realize I didn't like management and thus needed to switch back, the company wouldn't be in a particularly bad situation.
That being said, I feel the opposite about internal hires. One of the great joys of being a manager is coaching and mentoring the next generation of managers and having a healthy pipeline of candidates serves your company well.