If e was discovered to answer the question "what is the maximum of the real valued function x^(1/x)", sure, maybe that had some important application and was an important question in this hypothetical universe, but that does not make it a good definition.
"The number a such that d/dx a^x = a^x" says the same thing denotatively as "the number a such that when 1/x is integrated from 1 to a, the result is 1" except the former is more concise, and requires less to understand.
Connotatively, saying that e is "a such that d/dx a^x = a^x" immediately tells me where e is going to be involved, and its significance. Clearly, e is going to make an appearance in situations where growth or decay is proportional to current value. It isn't even a leap, or a derivation, it is plainly stated.
Saying "the number a such that when 1/x is integrated from 1 to a, the result is 1" does not allow me to immediately see where e would appear. Showing me that, I could not see its significance in application immediately, I would only know it as an interesting number with a relationship to 1/x.