This is a bit confusing. My understanding of how music publishing works is that the copyright is usually assigned to the publisher (for a limited amount of time

This is a bit confusing. My understanding of how music publishing works is that the copyright is usually assigned to the publisher (for a limited amount of time), so any rights of the copyright holder are rights of the publisher. They wouldn't have to explicitly assign litigation rights to the publisher. Disclaimer: IANAL, and with big artists, publishing deals can be very very different from boilerplate.