Because of this, typical out-of-office replies will often split the difference between our hopes and fears and say something like: “I’m out of the office and can be reached but if you need assistance right away, please contact x person.”
Found it! (Reddit’s own search functionality is garbage, but it’s so easy to google for reddit posts, thankfully)

For specific assistance, I’ll be responding to emails on [date]. If you need something resolved urgently, please contact [Contact Name] at [contact email].
I think the problem is that “at your earliest convenience” is a formulaic convention that uses explicit, almost exaggerated politeness to basically issue a stern direction, meaning “as soon as you possibly can”. When you turn it into “at my earliest convenience” it’s unclear if you mean “whenever it’s convenient for me to get to it” (what the words say) or “as soon as I possibly can” (what the meaning of the formulaic original is). Or else it sounds like you didn’t quite understand how “at your earliest convenience” works.