For reference, we had software running on the cloud, mobile, and devices, with multiple teams at each tier.
API changes were a mess of coordination. It wasn't "send JSON" it was "round trip this in a way that can be consumed by a device with 256KB of RAM." Multiple transforms of data across multiple mobile platforms meant just locking down on field orders could be problematic if someone didn't take proper notes, or if a Java programmer on Android didn't understand how to properly consume an unsigned 8bit int.
Early on in the project, we had a design team that hadn't yet been trained on what 96mhz CPU could do in terms of UI. I was doing careful reviews of UI designs, and eventually was just sitting in design meetings, to ensure that by the time specs got to my team the spec was physically doable.
Another example, when it comes to performance. I was going over EE schematics to ensure there was sufficient bandwidth to push pixels how the UX team wanted. Part of my job was working as a go-between for EE and UX and explaining the other teams POV to people with very different backgrounds.
Then there is the more soft-skill stuff. Seeing months ahead of time that another team would be having problems in the future, and suggesting to one of my senior engineers that making some friends over there and getting up to speed on their code base ahead of time might be useful, so that in a few months when help is asked for he can jump in and assist.
Letting senior management down gently that their favorite feature is going to be cut.
Prioritizing what features we need to implement to show "progress" to upper management so that we can all stay employed. This often required guessing far ahead of time what we'd eventually be told to demo "next week", and having work start on it way ahead of time. These were very much "do or die" demos.
The feature work? The team sat in a pod and discussed technical implementations with each other. I trusted the people I had hired to do good work, that is why I had hired them. I had the huge benefit of having personally hired every member on the team, which is one of the things that made any of the above possible.
Then there is the actual team management stuff. Inviting engineers to the right meeting so that they had visibility, giving them a chance to speak up in front of senior leadership so that their name was known. Taking someone who I saw leadership potential in and helping them see it, and giving them opportunities to grow that potential.
Rotating user facing features around between engineers so that they all had something they could point to and say "I made that shiny thing!" Good for both morale and career visibility.
Preventing burn-out. Telling PMs that work was going to be delayed because the engineer best suited to doing it was going to be working 6 hour days on light duty because they just went through a 2 month crunch.
Ensuring senior leadership heard the name of the engineer(s) responsible when something went right, and only heard my name if something went wrong.
Managing is a lot of work, but I think your judgements about it are unfair. Managing engineering projects is a blend of technical and people skills, and there is no law of the universe that says someone can't enjoy both.
 Or ignored the spec sent out. Or wasn't aware there was a spec. Or we were told we had to have this working in a matter of days so there was no spec. Or there was a bug in the deserializer that ended up swapping two fields. That last one took awhile to figure out.