The compound-interest intro to e (the value of 1 dollar compounded continuously for a year at 100% interest), to me, has several useful advantages over different introductions that are more mathematically rich:

- It's elementary to the point that you can introduce it whenever you want.

- It automatically gives a sense of scale: larger than 2, but not by a lot.

- At least to me, it confers some sense of importance. You can get the sense that this number e has some deep connection to infinity and infinitesimal change and deserves further study even if you haven't seen calculus before.

- It directly suggests a way of calculating e, which "the base of the exponential function with derivative equal to itself" doesn't suggest as cleanly.

I don't know of any calculus course that relies on this definition for much: that's not its purpose. The goal is just to give students a fairly natural introduction to the constant before you show that e^x and ln x have their own unique properties that will be more useful for further manipulation.