I think you missed my point entirely. I did those things for 10-15 years, and they led nowhere. Either it's one reason or another from a current employer, or none of it counts with prospective employers all because I didn't already have a manager title.
Sorry, I did miss that. It is unfortunate your employers are not supportive of your career ambitions. I've watched many engineers become managers at PlanGrid, so I know it is possible :)
Get good at your job. Get good at the company. Learn the domain and drink the koolaid. And, when the team grows, and they're looking for managers, nominate yourself to begin transitioning. You'll do both jobs and straddle the line for a while but eventually you'll have job experience managing a team.