Hahaha. This sounds like somebody thought the phrase “at X’s earliest convenience” sounded vaguely businessy and professional, but didn’t realize the pronoun is always supposed to be “your”. It’s never “my”, for the reason you mentioned.
My personal voicemail says that I prefer emails or texts when possible, and please leave a voicemail only as a last resort but either of the other two options will have a much faster response rate. I think I’ve gotten two voicemails in the last year.

Later that same year, on October 8th, President Dwight D. Eisenhower issued the first "Veterans Day Proclamation" which stated: "In order to insure proper and widespread observance of this anniversary, all veterans, all veterans' organizations, and the entire citizenry will wish to join hands in the common purpose. Toward this end, I am designating the Administrator of Veterans' Affairs as Chairman of a Veterans Day National Committee, which shall include such other persons as the Chairman may select, and which will coordinate at the national level necessary planning for the observance. I am also requesting the heads of all departments and agencies of the Executive branch of the Government to assist the National Committee in every way possible."
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.