I won't explain too much because my job is pretty niche and I don't want to get doxxed, but I've been promoted to "manager" recently. It's not the kind of manager you're probably thinking of like you'd see at fast food or retail places, I don't determine anyone's hours or pay or anything like that, but I am in charge of assigning work to the people under me and training new employees. The problem is I have to balance the work I do myself and the work I assign to other people. If I keep too much for myself then my bosses wonder why they keep the others around and they fire the peons and I get stuck with all the work. If I assign too much to the people below me then they wonder why I'm even around and I get put at risk of being fired. Obviously I want to keep my job because I have bills and shit but I also don't want to be the person who sits around all day and bosses the people who actually do the work around. I'm kind of just looking for advice.
You wear two hats. One hat is team-facing, the other is management-facing. To the team, you are the team lead and keep things running smoothly. The team doesn't hear management's stupid ideas or pressures unless they are a risk to folks' jobs or the projects overall. When facing management, it's 100% about bullshitting on performance and making sure those people feel good and are happy when you're talking to them. Underpromisr. Overdeliver. Hype up your team's work. You can balance the you vs. your team impression purely in terms of impressions and how you sell work that is already done.
In terms of balance between you and your team, this depends a lot on how brainwormed the rest of management is. Some places have a somewhat healthy mindset that if you're doing your job, your team is doing 90%+ of the direct work on deliverables. Other places promote narcissism among management by creating an environment where you only succeed if you can personally take credit. Your only way to sus this out is to talk to other people in management all the time and make "work buddies" with people that will feed you information. Go to the stupid management party at the bar or whatever. Have people over for dinner. It's terrible, but this strategy is basically just dealing with the reality of work being a petty fiefdom and you've gotta manage your place in court undet the feudal lord. The reality of leading a team well and what you present to management are two separate worlds with a bridge between them. When in doubt, assume that you need to do self-promotion, including promoting your team. Literally say, "[X important thing] wouldn't have been possible if X, Y, Z people didn't jump on it like they did. Our team really kerps this place productive. Look at how we made line go up." If you tell people what to think, they will often believe you.
In terms of being cool and not just being a tool for capital, the best thing you can do is lobby for more funding + pay + better working conditions for your group (call it "being competitive" and "retaining talent", which only works if your team is perceived as generating value). The next best thing you can do is shield them from layoffs by lobbying for your team being essential and by giving your team early info so they can start looking for other jobs (this will backfire if someone rats you out btw).
This is all really good practical advise right here. I don't generally lead teams as I'm usually hired as a consultant / niche expert, but I work with a lot of teams and the better ones use a lot of this. The terrible ones, the opposite.