I've been in the industry for over a decade, being a senior engineer, promoted to a tech lead for a short time, and now I'm a staff engineer at another company. As a staff engineer, I help multiple teams deliver on time with reasonable quality, I partner with product, and I mentor engineers, and I get to do little coding. I am going broad in several areas, but not deep in either of them: e.g. I do some architecture, but less than most senior engineers, I do coaching, but less than EMs.
I'm now at the point where I'd like to grow into an EM role, as I'd like to both have a bigger impact, and I find managing people and technology together, to be interesting. Unfortunately, my manager isn't much of a help in mentoring me, and the organization is quite chaotic. I feel stuck in my career: my organization doesn't have a clear career path beyond the senior engineer level, so I'm currently working in a kind void. I'm now unsure how to move forward. Forward meaning towards an EM role - or at least, how to figure out if this is for me.
It's great that you recognize you're stuck, and are starting to put the finger on why you think it is. However, it's still unclear to me why you're thinking that an EM role is where you would be happy, beyond this being an area you've not done before. Are you looking forward to spend most of your time helping people? Would you be fine being even more hands-off than you currently already are? Are you ready to deal with people when things go south, and have those (very) hard conversations? Are you okay with 17 other reasons being a manager can suck - even if it's the same salary, and less job security? Let's get back to the EM part in a bit.
In general, if you feel stuck in your career and want to move forward, your options are to do so within the company, or outside the company. If you want to get "unstuck" within the company, you need your manager's support and potentially your skip manager's as well. First, you need to map out what the options are within the org, if you're thinking of a team or a role change. While you can do this without your manager supporting you, think twice if this is a good idea. Even if you find a team that looks great, that team will reach out to your manager for feedback on your work.
What is your relationship with your manager, trust-wise? Are you open with them, on your struggles - even on your career struggles? Do you understand their priorities and how you can help them? Are you helping them? The same goes for your skip level. How often do you talk with your skip level? Do you talk about your career options? If the answer is no to either: you need to fix this.
Now let's get back to the topic of engineering management. You want to make sure you have the right motivations to become a manager. You should assume the position has no financial upside - internal role changes rarely have. Also, assume that it comes with tying your hands in new ways - you'll have to work through others rather than do it yourself. Finally, assume that you'll start as a newbie, and likely be pretty bad at it in the beginning.
Before switching, make sure you test the waters by doing activities that managers do day-in, day-out. Make your team better. Pick up ungrateful tasks. Give credit to others, selflessly. Teach and mentor and delegate well. Give candid feedback to others: including your manager. Network with other managers within and outside the company. Do your homework in learning about management.
Moving into an EM role is by far the easiest to do within the current company: but it still requires an opportunity and strong support from your management. This is why building trust with your manager and skip are so important: should there be an opening, they need to support you, often convincing stakeholders why you're a better fit than hiring someone else externally. And if your company is not growing and management positions aren't freeing up, you probably won't have an opportunity to move.
Moving to another fast-growth company, as an engineer, might be an option to help with a future transition. You likely won't be able to move as a manager, as it is extremely rare to hire someone into management, who has not been promoted to this role before, at a previous company. I've only seen this happen when moving from a well-known company to a small one that struggled to hire any leaders with past experience. The downside of this kind of transition - even if you pull it off - is you won't have mentors around you after you make the transition.
If your primary goal, however, is to move into engineering management, you might be setting yourself up for disappointment. You won't be able to control what opportunities come or how your team and company grows or shrinks. I personally never aimed to become a manager: I always looked for ways on how to help my team, and how to keep growing, and kept doing activities that are just as important for senior engineers, as they are for managers. At Skyscanner, I became the tech-lead-manager of the mobile team, both coding, hiring, and leading the mobile team. Then, I moved to Uber as a senior engineer. Here, following high growth, we were desperate to hire a manager for my team. I told my manager I'm interested in taking this role. I was already the team lead, doing most of the activities that a manager would do. My manager supported this transition and backed me getting this opportunity over hiring someone else externally.
Do consider organizations where there are parallel management-engineer career tracks. Your question touched on how there is no real career track defined beyond the senior engineer. This might be due to the size and scale of the organization - and it seems you might already be at the top of the not-very-well-defined staff level. However, better tech companies do have side-by-side career ladders with clear expectations for engineers and managers. If this was the case, would you still want to go into management - or would you be keen to make an organizational impact as an IC, with no direct reports? To give you an idea of career ladders that companies like Amazon, Facebook, Microsoft, Uber, and many others have, see the public ladders published by Monzo, Square, or Rent The Runway. My upcoming book will cover more details on what tech lead and staff positions mean at tech companies, expectations and career-growth wise.
The story of most engineering managers moving into management is similar: be a great tech lead on a fast-growing team, and opportunities to move into management come easier. Unfortunately, there is no single recipe on how to make this happen, or how to be guaranteed successful at it. Taking ownership of your career, getting your manager and management chain on your side, having allies across the organization, and working at a place that is growing quickly all increase your odds greatly. Taking on the responsibilities of the manager, without authority, and leading the team well is another good strategy. Just be patient and focus on the team succeeding, rather than on your putting your personal gain as first.
Finally, beyond certain seniority, there are no simple answers, and two paths are rarely the same. Good luck!