It is a natural mistake to make, so spending some time showing that it gives the wrong result is probably appropriate. Of course giving that wrong result a special name (especially something short like "e") or even calculating its value to a high degree of precision are pointless at this stage.
Then later when you have formally introduced sequences and how to prove convergence, you can show that (1+1/n)^n is monotonically increasing and bounded above, hence convergent. This is no longer a mistake, but closer to a fun (and quite difficult) mathematical puzzle than anything practical. Naming it "e" is still premature at this point.
Then even later when you've introduced differentiation, it's time to talk about the derivative of arbitrary exponential functions, which is where that sequence reappears, and giving e a special name finally becomes appropriate.
It seems like American math curricula are typically so excited to talk about e that they try to skip over all the intermediate steps?
The sequence (1+1/n)^n can be seen as a natural thing to study (indeed, tautologically, because we know that e is a natural thing to study) by various viewpoints.
The important aspect however is that the compound interest interpretation of this sequence is not only unnatural but wrong, in the sense that it starts from the wrong guess that a rate r compounded n times should be something like a rate r/n. It is unnecessarily confusing to teach such young students historical mistakes: bad students won't understand the mistake and good students will be bewildered. In both cases students come away thinking of e as a sort of "mystical" thing.
If I had to teach the compound interest, one starts by considering with a yearly interest rate r, your principal grows after m years by (1+r)^m. Now, can we find an equivalent monthly interest rate r2? To do so, we must solve (1+r2)^12 = (1+r), which requires logarithms, at which point you note immediately that r2 =/= r/12, which is perhaps unexpected. Now, it might be natural to ask what about for continuous compounding? Then, we study (1+f(n))^n as n goes to infinity, where f(n) is some function of n. We know it should decrease for large n, and the binomial expansion lets us guess (for integer n) that it should be f(n) = c/n for some constant c. Now, we are ready to compute the value of this limit.
The important part is that in studying compound interest one needs certain analytical tools, such as logarithms and asymptotic analysis, beforehand. It does not make much sense, as you say, to skip over all the intermediate steps and introduce the constant e as the solution to some unmotivated and unnatural formula relating to compound interest, but this is indeed what American schools do. In my experience, very few American high school students understand or remember (if they were ever taught it) the identity a^x = e^(x * ln a), and the concept of exponentiation is generally not well understood.
> To do so, we must solve (1+r2)^12 = (1+r), which requires logarithms
No, it doesn't. r_2 = (1 + r)^{1/12} - 1. Compound interest looks like (money) = (money_0)*r^t; you'd only need logarithms if you were trying to solve for time.
> Now, it might be natural to ask what about for continuous compounding?
I can't tell what you're getting at here. Once you've written down the equation (1 + r_2)^{12} = (1 + r), you've already provided a complete solution for continuous compounding. If the time you want to compound over is t, and Y is one year, then the solution is always given by (1 + r_2) = (1 + r)^{t/Y}. Nothing goes to infinity.
Well, I don't have my calculus textbook to hand, but I can tell you what I took from the class.
1. e is the exponential base for which f'(x) = f(x).
2. ln is the logarithm base e, and when f(x) = ln x, f'(x) = 1/x.
3. e is the sum of the series x^n / n! .
4. The textbook did specifically cover the fact that e is the limit of (1 + 1/n)^n as n goes to infinity, and it also specifically tied this in to the idea of computing interest by an obviously incorrect method. You could only call this a "natural mistake to make" in the same sense that it's "natural" to assume the square root of 10 must be 5, or that the geometric mean of two numbers is necessarily equal to the arithmetic mean.
5. However, the limit is important in that it illustrates that one to an infinite power is an indeterminate form.
6. As detailed in points (1) and (2), and hinted by the name "natural logarithm", we measure exponentials and logarithms by reference to e for the same reason we measure angles in radians.
It's possible that this particular definition of e is important to a proof of one of the properties of e^x or ln x, but if so I don't remember reading about it in the textbook and it wouldn't have been covered in class. In my real analysis class, we used the Maclaurin series for e; (1 + 1/n)^n was never mentioned.
(It's really easy to show that that series is monotonically increasing.)
> Then later when you have formally introduced sequences and how to prove convergence
This is not material you'd expect at all in a calculus class. If sequences are mentioned, it would only be in passing as you move to series. Several methods of testing infinite series for convergence are covered. What it means for a sequence to converge is not. Limits are not defined in terms of sequences. Infinite series would be covered after, not before, differential and integral calculus.
You have to have a name for e because otherwise it would be impossible to work with. But it is interesting and the wrong way to compute interest isn't; there's no point in trying to motivate something important with something unimportant.