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.
Crowd shot from Labor Day Rally | Construction & General …. Examples of out of office messages for holidays. If you have any pressing questions, please include the word urgent in your subject line so i can make your email my top priority during. I am out of the office for the holiday weekend, however, i am responding to emails that need immediate action. I'm out of the office until date.

Pet peeve: a fe people I know use the OOO reply to say something like ‘Have a great day!’ It is on all the time, and for no helpful reason.
I’d side-eye if an OOO for a couple of days off said that, unless someone was in an unusually time-sensitive role. But if someone’s out for long enough that there’s an alternate contact provided, that’s long enough for them not to be wading through missed emails when they get back.