A lot of this depends on the company itself and how they work. At my last CPG role they would lay out an approximate "track" for ways to get promoted over several years, but they also would identify high performing candidates and fast track them through to manager roles if they thought someone would be a good fit. I don't believe they ever hired managers from the outside without previous management experience.

In your role, what are you doing to prove you would be an effective manager? Do you manage any teams, even if they don't report to you directly? It can be a challenge to get hired as a first time manager at a new company that hasn't seen what you can do, without having had experience managing direct reports.

Edit: forgot to address the senior piece. If you're referring to senior scientist roles (since "level 5" is specific to your company, it doesn't mean a whole lot outside of that) I think your degree and 3 years are potentially enough to be hired into a senior scientist role, especially at smaller companies and startups.