The bash man page says:

Hello,

I have just spent a considerable amount of time finding out where the
hell $PS1 gets set when I use --rcfile, which wouldn't have been
necessary if this was documented properly. So please fix this.

Thanks,
Christian Ohm

I don't think the documentation is the one that should be fixed. What is wrong is the current behavior of executing the wystem-wide even if the user requests a custom --rcfile.

Currently, there's no way for a user to prevent the system bashrc to execute and still use a custom bashrc.

A patch was sent to #516152 and upstream. If accepted, both bugs can be closed.

Cheers,
Rodrigo

For the record, this issue is now considered fixed upstream after a
change in documentation:
https://git.savannah.gnu.org/cgit/bash.git/commit/?h=devel&id=a91b8b077300b0a2a7daefe02f0363f9116e00d5

That commit will be part of Bash 5.3.