While this could be implemented in core, there is a fair amount of customization people might expect (e.g. formatting or rotation policy), and then there is the issue of how to display the recorded logs.
I am thinking this functionality might better live in the Support Core plugin: either at your option or by default, it could write custom log recorders to $JENKINS_HOME/logs/custom/$name.log (using the nice compact SupportLogFormatter), and then those files if present included in the support bundle.
Audit Trail plugin does something similar, but only for its own logger.