I’ve run into the “no voicemail” thing at a few businesses where phone was the main mode of contact too, and it was hugely frustrating. You call your doctor to ask about, say, a billing issue, and it turns out they’re closed, but then it just says the office hours and “goodbye *click*”. Seriously? Sorry, /end rant.
To my mind, it’s just a really bad idea to mass delete stuff that comes in while you’re out. You never know if those emails contain important (though non-actionable) information that you’ll need. You really can’t expect people to resend information like that once you’re back, especially if you were cc’d on something.

Hi, I will be out of the office starting [MM/DD] through [MM/DD]. If you need immediate assistance during my absence, please contact [name] at [email] or [phone]. I will respond to your emails as soon as possible upon my return on [MM/DD].
The attendant menu for your main contact number may be the first form of contact that a customer has with your business. Make sure your greeting is clearly articulated. Provide an option to speak with an operator if the caller has a question that is not addressed by the menu options.