Incidentally this strategy is why I worked at startups before this job. One did land me a lead position with 1 report, but it was when the company was falling apart and thus not the most meaningful experience. Another took the role I'd been doing by myself, and replaced it with a team and a newly minted manager... who wasn't me. I got shuffled under someone else who didn't want to manage and promptly quit, so that didn't work out either.
My company theoretically prefers senior ICs internally sourced for EM roles - it's the only place I've been able to interview for EM positions, and I very nearly got one - the hiring manager said he was set to pick me, but then they decided they really wanted somebody with deep experience at the last second.
Actually, interviewing for an internal transfer at a big company has been the closest I've come. I've also found that the big company is a lot more inclusive; startups tend to be cliquish and if you're not part of the boss's in group you will really struggle. Here, I can get involved with many things and many teams directly, so even if my boss doesn't have a job for me, I can work the network and find my place.
At least... that's the strategy I'm trying now. "We're a growing team and that position will totally be there for you a year or two down the road" applied to 4 jobs over almost 10 years without yielding any fruit. (and 2 of those companies imploded, so I didn't even get RSUs like I get now)