Make following exercise: how many managers you know are actually spending much time with "helping people solve their problems" or "patiently answering dumb questions"? Managers I know spend some time with it, but frankly, much less then analysts and not much more developers.
Another exercise: Do you know managers that are unpleasant around, arrogant or misinterpreting people? Do you know managers who are actually not good at all at communicating down chain? Because I know a lot.
what the exercises show is that none of that pleasant communicating to all has to do with who is actually successful as manager or selected for management.
Yet, politics is not the same thing as managing.
Toxic politics are everywhere, but the ability to negotiate, navigate the organization, find what make people w/ authority tick, is crucial to any successful manager.
Politics is how every organization makes decisions. It's understanding the information flow, the decision making process, people incentives, and how to influence these to get the outcome you want.
Managing up is about altering decisions made above your level (info moving up) and managing down is about taking those decisions made above your level translating them into decisions that affect your team.
i.e. Managing is very much about politics.
Engineering has politics too, just different kind of politics. You also spend a lot less time explicitly doing politics. But, a lot of what is going on in code reviews for example are dominance games and status managing. How you present yourself so that they perceive you as technical the right way matters too.
If that has not happened to you even after being clear, it can mean different things. It could be that you are within a group that has little growth, and does not need a new EM. Some political reasons. Maybe your manager is terrible and does not seek opportunities for you. etc...
In this case, my advice would be to join a mid size company which has growth, and you join as a senior engineer. Startups can be another way (that's how I did it), but that's more difficult to plan. I certainly did not plan it this way myself.
I am aggressively hiring EMs, and will definitely consider first time managers.
From that and your whole post, I'd take a guess that you are a top contributor easily getting more and bigger projects.
My advise would be to stop getting more responsibilities. It has to be done by your end, you need to not do anything more.
There's some more work coming up? Don't take it, don't get involved in email threads about the new projects/tasks. They will find somebody else to do the work. They want you to be oncall after hours? Don't do it, phone is off.
Good fast workers are a magnet attracting more and more work until they burn out under the load. Look around you and notice people doing no more than they are required to, learn from them, it's the only way to last more than a few years at any place.
I really like the comment about "would you do it for the same money". If the answer is no, you are probably better off not doing it.
This isn't the first time I've been in this or seen this situation play out. Businesses stacking an impossible list of responsibilities on one person until they burn out and leave, then get replaced by a manager and 2-3 others.
So I ask, for those of you who have intentionally made the move, how did you do it? I feel like I can't be any more transparent with the business that it's the direction I'm looking for and that I will leave soon if it doesn't happen. But then what, do I keep playing this game into perpetuity until my luck changes?