aren't the system prompt of Claude public in the doc at https://platform.claude.com/docs/en/release-notes/system-pro... ?

The system prompt of claude code changes constantly. I use this site to see what has changed between versions: https://cchistory.mariozechner.at/

It is a bit weird why anthropic doesn't make that available more openly. Depending on your preferences there is stuff in the default system prompt that you may want to change.

I personally have a list of phrases that I patch out from the system prompt after each update by running sed on cc's main.js

This is for Claude Code, not just Claude.