Writing is a strong skill - being able to articulate your ideas, whether in a design doc or other communication is critical. Clearly explaining pros and cons and whatever alternatives you considered - I wouldn’t discount any of this work.
I was at one point on a team where I didn’t feel like the technical side was sufficient (for me), and so I left that team.
I wouldn’t let my ability to quickly grasp some code base on github be a personal test for whether my skills are rusty or not rusty. That sounds impulsive to me (no offense but I would react this way earlier on in my career).
Try to tease apart specifically what it is you’re missing - do you not feel fulfilled? Is there a certain kind of technical challenge you want? Is there a specific problem domain you want to work on? I’d start with that to figure out what it is you want to do.
It's fine IMO. Work is about producing value for your employer, so do whatever it takes to do that. Usually it's about meetings, which are pretty much therapy for management.
If you want to be better at code and can't work it into your day job, then you'll have to work on projects outside of work.
Also consider taking it a step farther, and go ahead and initiate some of those meetings or email threads. There's a reason people are making them. They help the team stay in sync and feel better about the project.