Upload a new version of your paper on arxiv, this time with an email address that works.
It is a good idea in general to make sure that your papers contain up-to-date contact information. One way of doing this is to use an orc-id.
Who sends emails to paper authors? How often do they respond? How fast do the email addresses go out of date? I lost access to my email address included in most my papers within 2 years of publication.
I see little to no value to have it included in the paper.
Also, I don't think we are yet at the point when human2human communication is not possible.
I do when the paper is not easily available or the publisher charges some outrageous fee (have seen $50 for a paper in the past).
Authors typically despise the publishers and are happy to share their work to anybody interested.
I do, when I'd like to read a paper that's locked behind a paywall and not available on sci-hub. Authors of scientific papers are much like any other authors... they want to be read. The more enlightened among them understand that obscurity is a problem rather than a perk. They also tend to appreciate engagement in the form of follow-up questions (at least from people who actually read the paper.)
Obviously it's not a major concern on arxiv, but in a larger historical sense, this type of communication was a key original application of email.