As with many "we should explain core math things better," there is a 3Blue1Brown explainer that attempts to give intuition for e:

https://www.3blue1brown.com/lessons/eulers-number

As suggested by the OP, it approaches the problem from the angle of showing that e^x is the only function that is its own derivative.

There is also a follow up explainer giving intuition for e^ix as being about modeling rotations.

https://www.3blue1brown.com/lessons/eulers-formula-dynamical...