I've entered 2 new orgs as an engineering manager, after getting some experience organically. You need to prioritize between lots of things, including technical / people. Usually people is the right choice I've concluded, but the first time my teams were using all new tech for me and I really struggled after over focusing on relationships. The second time I still focused more on people but new the tech better, and forced myself to find time to go through more typical developer onboarding. It was way more successful.
Some of the things you mention sound right for a team lead (i.e. single team) but not really for an EM where you might have 2+ teams. You need to be able to solve the problems the leaving teammates create for you, but stepping into the role is probably the last thing you should do. Don't get me wrong, you need to live the life to build credibility and empathy, but doing the job yourself is usually a substandard, unsustainable solution.
Some of the things you mention sound right for a team lead (i.e. single team) but not really for an EM where you might have 2+ teams. You need to be able to solve the problems the leaving teammates create for you, but stepping into the role is probably the last thing you should do. Don't get me wrong, you need to live the life to build credibility and empathy, but doing the job yourself is usually a substandard, unsustainable solution.