My husband’s voice mails says “…if you need immediate assistance call Mary at ####…”, only Mary retired something like eight years ago. I mention this to him every once in a while. It hasn’t changed.
Yes, mine (for external e-mails) typically says somethingalong the lines of “I’m out of the office until [date/time] and messages to this address are not monitored in my absence. I will respond as soon as possible on my return. If your message is urgent, please re-send to my assistant [email address] or telephone [assistant’s number]” Internal it will usually just say “I’m out until [date/time] pass any urgent enquiries to [co-workers] ” although my assistant and a couple of others have my home number and personal e-mail so can get hold of me in a genuine emergency, and I will sometimes speak to them in advance if I’m willing to lower the bar on what amounts to an emergency, but I would not expect any of my employees to do that !

Before I implemented this system, I got *way* too many calls asking if so-and-so was out of the office that day when an OOO tells them that Bob will be back in three hours and to call Sue in the interim. The OOOs have reduced status check calls to practically zero, and the OOO preview pops up in our system as soon someone’s email is entered.
No one thinks much about cybersecurity while traveling. However, email security company Tessian warns the out of office message actually plays right into the hands of threat actors and cybercriminals. It’s a social engineering attack vector that no one thinks about. The out of office message is ubiquitous and handy. But if it includes any personal information at all — such as attending a funeral or going out of the country — attackers have all the information they need to impersonate the person who is out of the office, without the attacker having to do any real work.