Give complete details of the person they can contact if there is an urgency. If you are in charge of multiple departments, you can add more people along with their expertise area so that customers know who they should reach.
That really does feel excessive. I think most people are savvy enough to think, ‘Huh, it’s late in the day for pretzelgirl, I might not get a response until tomorrow. Business as usual.’ Being out of communication for a day – travel, meetings, days off – doesn’t require an out of the office alert, either, just a response as soon as possible.

I find it rude, as well. If someone is emailing you (the royal you), it’s because they need something. Saying you’re just going to delete it without also giving that person a Plan B contact is totally rude. It sucks having to go through emails, it does. But it’s part of the job. In my role, I get requests from internal colleagues and from external partners. Even though I provide those partners with an alternate email to send their requests, it’s still my responsibility to make sure that the requests that were sent to me in my absence were handled. Saying “everything I’m sent will be deleted” just Would Not Fly in my industry.
Hi, Our store will be closed until the end of the week for Thanksgiving/Christmas/New Year]. We are happy to inform you that all of your emails will be answered once we are back on [date]. Kind regards.