I just can’t see how things like Claude Code are not keeping your generated code for other purposes … Should developers care more about this?

No. I don't care.

But I sympathise with those who do care.

I'm not sure why it doesn't bother me. Maybe it's because I see myself as a problem solver rather than a coder. Coding is just a means to an end. The LLM is just another tool. My code is less important than my ability to solve problems.