I'm curious if you discussed your desire to move into a management position with your supervisor. If you did discuss it, what, if any, feedback did you get?
Based on your statements in your question, the frustration is obvious, so is a sense of entitlement, like you've earned this right but it's not happening for you.
I am sorry that you are feeling like that, it's a terrible place to be and it can drive you to make some really stupid decisions. Hopefully, in simply asking this here has helped you relieve some of the stress, if not, maybe the following will be of some help.
This is not a direct answer, but should give you some ideas and thoughts to ponder, plus a couple of next steps.
Here are some of the themes I'm seeing in your question:
* Too much stuff to learn when it comes to coding
* Want to spend more time on personal projects instead of on technology treadmill
* Feel like you can complete coding tasks quickly
Let's address these themes:
I'm going to treat leadership / management interchangeably here, even though there are some serious differences between the two roles. For the purposes of being a line manager, they can be treated interchangeably.
Moving into a leadership role, whether you're looking for a tech lead or a management role, does not stop the first bullet point above. If anything, you have to be aware of all of the new trends enough to make a judgement call on it's validity, maturity and use-cases. You need to know this because you'll have your reports coming to you on a regular basis wanting to use some of these techniques / tools / languages / etc. How do you say no without offending your report? On the flip side, you'll have your manager(s) coming to you asking if something is a good idea. How do you explain the tradeoffs if you're a one-trick pony?
To the second bullet: you have, significantly, more time as an individual contributor than as a manager. As a manager, you have to make sure everyone is on target with what they need to do, run 1-1s to make sure the people are satisfied / happy and are not planning to resign shortly, report up the progress, report down the needs of the bosses. You are, perpetually, stuck between trying to communicate up, down and across the organization, while sitting in a lot of meetings (assuming a large org) and being on the hook to get more stuff done. This is doubly so if you are new to management. The hard truth is that the role doesn't matter, you have to know where to draw the line(s) and have a life. It's a constant tradeoff, do you pick up a new hobby or go out with your friends or learn the latest framework / language / algorithm / etc.
Your ability to code or get engineering tasks done quickly has very little to do with your ability to lead / manage. You need to learn new skills, mostly around communication, relationship building, messaging, ownership, delegation, accountability of others, etc. Just because you are a good developer does not mean you'll be a good manager. I would argue that most developers should never be allowed to lead as the skills and roles are so different, they are not transferrable. This means that you're starting from scratch, again. By the way, as you get promoted, you're constantly starting from scratch, unless you are, already, doing some of the things your manager is doing.
None of this directly answers your question, as there is no direct answer or how-to guide, it is very industry- and organization-specific.
Being a manager at a startup is very very different from being a manager at a large corporation. Moving into that role is, also, a different process. In some ways, moving into a leadership role at a startup has more to do with raw leadership skills then anything else; you see a problem, you own solving it - rinse and repeat, boom you're a leader now. By the way, replace __boom__ with months, potentially.
In a larger company, it's more involved as managers are expensive, both in terms of salary and training / knowledge acquisition. Imagine the disruption to the team, the delivery flow, unofficial communication channels (aka social glue), etc. when a manager leaves.
To be perfectly blunt, if you have ask the question, you need to really think about what it means to be a manager / leader. Look at how your role models came to be leaders, what paths did they take? Based on what you stated, your goals are position and money, that will make you a terrible manager, if you don't care about the people. You'll have a hard time retaining people or producing results in any high-performing organization as you won't be able to motivate anyone based on those drivers.
Another thought, position is meaningless in a high-quality tech environment. Power does not come, just, from the title, it comes from the group differing to you because of respect. Having worked at places where the position/title alone denoted power, it's a crappy place to be.
My suggestion, as to the next steps, is to:
* Reflect on the instances when you were put in charge of something and how people reacted to you, how they treated you and how you treated them
* Have several coffee dates with managers / leaders in your organization (don't go outside of your org) You are looking for people doing the role you would move into as your next step and their managers. Tell them you're interested in being a manager and what lessons they learned, what their day looks like, what keeps them awake at night, etc. You want to learn as much as you can from them and get your name out there. This is part of building relationships but also information gathering. This will tell you if you actually want the job and are willing to put up with the politics and dealing with people without the context of code to help you.
* Sit down with your manager in your next 1-1 and explain what you're seeking and ask for help to move into that role.
Also, read Managing Humans by Michael Lopp. It will give you a good idea about what it's like to manage engineers.