In high school we were taught that it is because The slope of the function e^x is equal to e^x. Growth being proportional to magnitude is natural.

Nonsense, this equation doesn't even make any sense without a well defined notion of the exponential function, and then a well defined extension of said function into the complex numbers. You will already have e by the time you reach this equation because defining what exp(z) even means requires you already know the properties of e^x over the reals you wish to preserve. And Euler's formula comes from finding such a function and then defining it to be exp(z). Multiplication in the complex plane by a unit vector is a rotation. Exponentials "repeated multiplication" by such a vector is spinning. And it turns out spinning at a constant rate satisfies the properties of the exponential function so it makes sense to say that's what exp(ix) means.

This is perhaps the most unnatural equation (well identity) in maths. It doesn't fall out anywhere, you would never write it down and solve for e, it's a special case of a more general result you would get first, and it's symbol soup for precisely the reason that the identity itself confers no understanding.

exp/log are natural because you almost can't help but discover them as they appear in so many different seemingly unrelated places.

Not really. Your equation has no unknowns, but if you consider the "e" in your equation as the unknown, and solve it (here I use "a" instead as the unknown, and use "e" in its usual meaning):

We want a^(i pi) + 1 = 0. Now,

a^(i pi) = e^(ln(a) i pi) = e^(i ln(a) pi) = cos( ln(a) pi) + i sin( ln(a) pi),

so we want cos( ln(a) pi) = -1, sin( ln(a) pi) = 0,

so ln(a) = 1, 3, 5, so a = e, e^3, e^5, ...

Thus indeed e^(i pi) + 1 = 0.

But also ln(a) = -1, -3, -5 work, so for example for

a = 1/e = 0.3678794412... > 0, we have

a^(i pi) + 1 = 0;

and of course for a = e^-99 = 1.0112214926104485... × 10^-43 etc.