I thought it was funny but could never get away with using something like that at my org. I loved the “competent people who work for me” part – I make this joke all the time. We have some people who feel that they should have a manager personally attend to them and, at least in my case, my highly competent team is in the weeds of that work a lot more and are not rusty (like I am).
Now make your email unique when you are out for traveling for work and be a standard part of the job. If you are traveling to a conference you can set up your email.

The main thing to equivocate over for this sort of response is what details to include as a backup contact method. If your line of work means clients might need to urgently contact you, a mobile number might be appropriate, or if you’re lucky enough to have an assistant (must be nice), you can list their contact details.
Then there was the occasional one who would do what Alison mentioned with the sickness excuses, and create a tale that read like a police report: “I must miss my deadline because, on the night of August 12, my 45-year-old sister was alone in her house when an intruder entered. He was a 6’1″ caucasian male wearing a black balaclava and carrying a candlestick. As my sister approached him, with the dog barking around her heels, she heard a distant car crash which led her to have a fatal … etc.” (This is not an actual excuse I received, just similar in detail to some of those that were submitted.) These ones I was pretty sure were a writing exercise, requiring time and effort that could have been put to better use on the actual assignment they had been given.