What they told you is correct. The Apple ID will continue to work regardless of whether or not the email account is still valid, but if you want to receive the email notifications you need to change the email address being used as your Apple ID to a functional one.
You are not changing your Apple ID, you are updating the email address it is using. Updating the email address for the Apple ID does not change the Apple ID in and of itself. It retains all its purchases, and access to all its subscriptions and services, but provides a new functional email address, so notification emails can be received by you.
Changing the email address in the system, as in what account is set in the mail app or somewhere else, does not update your Apple ID.
If you wish to leave the email address as is for the Apple ID, then you can, but you won't receive notification email once it stops working.
The Apple ID will continue to work regardless. But there is really no need to keep a non functional email address attached to an Apple ID.