1) Mathematics majors should be taught how to prove. 2) Engineering/Applied Math majors should be taught to use.
Fortunately or unfortunately they take the same classes. Personally, I think the obvious solution would have been to just take whichever classes you want to and to get "a degree" once you fill some sort of criterion. But the CAs and actuarial scientists, uhm basically everyone, don't want that human resources headache to actually read through someone's CV patiently and ask a few patient questions.
So you might drop out of either of the two by year two and so you don't have the full four years. [1]
[1] In South Africe, a bachelor's degree is three years. I don't know whether this is a good or bad idea, but that is how it works. Your fourth year is called "honours" and is a separate degree.
Most people just want to calculate a P value or a 95% confidence interval for the mean of whatever they’re researching. They’re not interested in how it all works.
Another child poster pointed out that in special cases (like the limiting case of the binomial distribution) you can get away with other arguments. And you can certainly 'prove' it via simulation. So maybe you don't need the full generality of Fourier analysis in the end.
I would call that a colloquial definition of understanding. Very different from mathematical understanding.
If you ask a mathematician what you’d need to know to understand Fermat’s last theorem, he won’t say “high school pre-algebra.” That’s only enough for you to understand the basic statement of the theorem. It doesn’t get you to the why. To understand the problem involves a deep dive into both algebraic number theory and analytic number theory.
For example, most mathematicians would say they understand large numbers of theorems that they cannot prove off the top of their heads. On the other hand they probably have what Borovik in "Mathematics under the microscope" called a "recovery procedure." That is, a set of constraints or path that reproduces the result. They know they can reconstruct the proof if they need to from the various gambits and skills they keep polished.
Also, the proof via the normal distribution being an attractive fixpoint of convolution is fine, but it only works on a particular subset of functions. We know the theorem applies beyond that subset, and there's a cottage industry of extending it in bits and pieces and calculating better convergence bounds. There is no proof available today that says central limit theorem applies iff conditions x, y, and z. So in this case the proof really can't be said to be understanding.
Now, that's well and good for probability alone, which is a field of mathematics. Statistics isn't a subfield of math, or is a subfield of math the way physics is. For a statistician, understanding the central limit theorem is much more about knowing what kind of observations it is reasonable to expect it to approximately apply to, what kind of tests rely on it and which don't, how to check if it applies in a rigorous way, what kind of visualizations and exploratory data analysis is enabled if it does, how the normal distribution and convergence to it fits into a whole family of distributions and features thereof...