If one were concerned about the message rubbing people the wrong way, it’s certainly something that can work with a bit of rephrasing: “I’m out on leave from X to Y. All messages received during that time will be deleted. Please resend your request after Y, or else contact Z for urgent assistance.”
There is no one right day (or way!) to send a letter to a customer. However, holidays and special occasions can be a great opportunity for your business to reach out. Holiday letters are an excellent way to keep customers up-to-date, send out promotions, and show customers you care.

Voice mail is a strictly worse medium than almost any alternative. You don’t get the opportunity to converse and ask questions back and forth like you do with a phone call; you can’t extract information efficiently from the message like you can with an email (the old “ugh, I have to listen to this entire message over again just to check one thing he said at the end” scenario).
Yes! I remember reading here the phrase: the default mode of clever is asshole. Meaning when trying to be clever backfires, you end up just looking like an asshole. I’ve given up the need to get laughs at my clever sense of humor while at work. I really hope that out of office message is for internal emails only, because the risk of this landing badly is too high.