This scenario is far more common outside of the tech industry, where you usually have a CIO running the IT cost center, and a CTO making decisions about technology adoption strategy.
I once worked for a company with a self taught CTO (and not the good kind). They had a number of star players, and this CTO would frequently lash out at them. All because he was getting in the way of them doing their jobs, doing work he wasn't qualified to do, trying forcing them to clean up after him, and then yelling at them for it. It was insanely toxic. I only lasted a few months. It was so bad I back channelled patches and project briefs to people he liked to get them approved.
Had this CTO remained people, project and product focused everything would have been fine.
Was that a Fintech in Germany, by any chance? :)
I once witnessed a meeting between a CTO and a Tech Lead. The CTO was attending from his laptop in an open office, and he was yelling in Russian for one hour straight at another Tech Lead because he wanted the tech lead to finish his work. It was a pathetic display, with the whole company watching and wondering what was going on.
Eventually he was "phased out" by having a few people promoted to VP of engineering who would deal directly with the CEO instead of him.
Last I heard he tried to rewrite the financial core in Golang by himself, but he failed since nobody wanted to work together with him and he doesn't really knew the language.
In most companies I've been a part off, including multiple >$1B tech companies, the CTO's focus is not on the engineering. That's the job of a VP Engineering or some similar position.
CTO (which will sometimes have a "CTO office") is to work besides the engineering on investigating new technologies and ideas that are beyond what the engineering organization would have otherwise done on the day to day. They are also an authority on all technology in the company but are not in the engineering "chain of command".
That said this is not universal, there are organizations where the CTO does lead the engineering organization. I think that's sub-optimal because there is always going to be tension between the day to day and the broader scope and those should be different roles.
In a startup, it is more common for a CTO to lead engineering because there is not yet enough to justify having both a VP Eng and a CTO and perhaps most of the work is around figuring out technologies. But as the company grows it makes sense to separate those functions.
When I was CTO of my startup I had far more direct engineering development work, but that is typical in the building stage.
As for the core of this post, the one thing I do agree with is the ability of the CTO to actually be technical. I write code all of the time, but not for our products. The goal is to remain both technically proficient but also focus that proficiency on leadership.