The item becomes unusable and gets the status ''broken''.
I assume this is not intended.

I have not yet tried if it can be fixed with mending enchant or the repair tool ritual from occultism.

What modpacks do with our items, we have no control over.
If it's not possible to fix fishing rods because of modpack limitations, then that's on them.

The fishing rods is indeed meant to not work once "broken", but can be fixed in an anvil with the material it's made of.