I've seen (1+1/n)^n before, but never seen an explanation of why I might ever want to use something of that form. I've used the e^ix notation extensively, but again I've never really cared, because to me it was just a compact representation of sin and cos together. Likewise, all the proofs in that article are still a bit "so at this point on this carefully chosen graph, the gradient is e" and I think "Who cares? You carefully chose the graph to prove a point, I'll never see it in the real world."

And then the example with compounding interest - immediately I can see the application. It's definitely a good way of explaining it, although maybe it'd be even more grounded if it had n=12 and n=365 as examples. When you notice that the actual values seem to be converging, then you can try plugging in ever bigger and bigger numbers. This way you can discover the value of e for yourself and that process of discovery leads to a better understanding than rote learning of an abstract thing you haven't mentally visualised yet. All the other explanations are useful later, and they allow you to see it in different situations, but having at least one "this is why it's tangibly useful" hook at the start is definitely a massive help in understanding something.