Are non-coding Team Lead positions common in California?

Question

Work organization

Colleagues, I am considering moving to the States, most likely to the Valley, Boston or New York, although the list may change. It so happened that at the current place of work for half a year or a year I have almost not written code, there is simply not enough time. Previously, when I was responsible for one team, I managed to write and review, now there are 2 teams (10-11 people), it is very rare to even review.
Everything I do, from holding daily meetings, translating business problems into a language understandable to developers, solving algorithmic problems (top-1% on hackerank.com, we have projects with good mathematics), holding demonstrations (and collecting requirements) for customers, interaction with designers, ergonomists, outsourcers, etc. Objectively, I feel that I really like such a routine (with a minimum of writing code), but on the other hand, I worry that I am losing my value as a playing coach. In other words, I wouldn’t want to write only code now, or write code most of the time, I’ve already eaten :)
Actually, in connection with a potential move, I had questions, I couldn’t give myself an unambiguous answer based on the vacancies I looked at:
1) Are Team Lead positions common in California, Boston and NY with no direct or minimal coding?
2) Perhaps such positions in the States are called something else than Team Lead? Actually, here my position is the Head of the Development Group or the Head of the Direction.
3) Is it worth it to return to writing code a little, thereby increasing your chances?
Thank you.

Answer 1

Team Lead writes code no less than he is smart. Usually difficult parts. Not all the time, but quite a lot.
Otherwise it's a different job.

Answer 2

You are not a Team Lead. You are some kind of PM + Manager. My advice from the youngest to the oldest, hire assistants and delegate distant and less important tasks to them, and keep the most delicious for yourself. And finally start coding.