It's straightforward to get yourself to a place where you can do cryptographic things and feel somewhat comfortable with what's happening. Truly understanding it to the point where you can reason safely about it is deceptively harder.
For example, the proof that there are infinitely many primes looks simple [0], but it's still pretty hard to understand, let alone derive yourself independently. And the other important cryptography/number theory theorems like Euler's totient theorem [1] are even trickier.
Along those lines, you do not need to understand the proof of Euler's totient theorem to understand cryptography. It is a distraction. All you need (at most) is to know that the result is true, and even then it's only fundamentally important for RSA, which you likely shouldn't bother learning about these days. RSA simultaneously
1. looks very simple (though the simple version is horrendously insecure), and 2. does not have particularly good performance, and 3. does not have particularly good security (either post or pre quantum), and 4. has been in the process of being phased out for quite some time now.
this is not a good combination of properties. The fact that cryptography textbooks cover it is mostly due to historical tradition. I would personally argue it is time to omit it from instruction materials.
Well, I had to when I learned cryptography, but I learned it from a class offered by the math department, so I guess that's rather unsurprising :).
> even then it's only fundamentally important for RSA […] this is not a good combination of properties
Strong agree here.