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.
Thankfully, there’s a way to respond swiftly to all incoming texts without needing to jump onto your computer or phone. The answer is simple—automated text messages.

In conclusion, an eager customer trying his or her best to reach out to you is the last person you want to disappoint. Make their day special with an unexpected or quirky autoresponder email that your customers wouldn’t have thought they’d see in their inbox in a million years.
Oh I also saw one from a person who used to be my manager (thank goodness that nightmare is over). She had: – An extra space in the email address to contact in her absence, which would create a bounce back if someone tried to use it as she typed it. – Had a date that was clearly a “fill in the blank” that she didn’t look at, because it was something like “3th” instead of “3rd.”