Also, get ready for a lot of gatekeeping. “Oh, you don’t really want to be a manager! It’s so much more responsibility and work. You should be happy to be a foot soldier for the rest of your career! Management is not all it’s cracked up to be. As you can see I’m stressed all the time! Now excuse me, I have to go close on my second vacation home in Hawaii.”
[EDIT] my suspicion here is that yeah, it's basically "luck into it". I mean that's how everything else I've done career- and pay-progression-wise has worked (luck into someone tasking me with something they definitely wouldn't hire me to do, but after I've done it for a few months someone else would) so maybe this'll work out the same way.
Is it even possible to find a management position outside without management experience?
In corporate jobs, this is rarely the way in. Most likely, you'll have to convince your current manager (and maybe theirs) that you're ready to be a manager and look for a manager opening elsewhere in the company. If you think you're ready and your current employer does not, you'll have to look outside.
Years of leadership seems like a purposefully vague credential. It's not about a title; if you don't think you've been a leader at work, you probably weren't (or maybe you were and your talent just wasn't managed properly ¯\_(ツ)_/¯ ).