Realize also that many engineers are disdainful of managers and management, so take any feedback you get with a grain of salt. Explicitly aiming to become a manager will get a bad reaction from people.
That said, managing people and process is a real set of skills at which people can be good or bad. You can practice and improve these skills like any others. There are plenty of people who are managers who are poor at it, but that's no different than any other job. And no, experience itself won't grow you the way you want. Adding side training will be essential, whether it is individual courses, Udemy and similar coursework, reading, or full degree programs.
I think an Master's in this "Management and Project Innovation" would be valuable. My path was realizing after about 7 years of software work that I really enjoy the larger activities of businesses even moreso than the software development, so I got an Executive MBA. That gave me some negotiation power to finally slot into a leadership role. It also rewired my brain and vastly expanded my knowledge of leadership and of understanding business concerns to be effective. It really revolutionized my whole way of thinking about the world and made me a better person in my personal and professional lives.
So my take would be: if your passion is in _orchestrating_ projects, do consider that MBA and angling for a Project Manageer role as well. But you should also get a year or several of software development so you learn the realities of people working on software projects. I think practical experience is important to stay grounded when you get with with management theories and practices. Without real-world experience anchoring you, it can turn you into a Pointy-Haired Boss personality.
If you already think that is going to be an interest to you, I encourage you to go work 2+ years as a software engineer, then start a Master's degree concurrently after you have some experience.
Email me if you want to talk in more detail.