Consider an 8-bit RNG.

By your argument, it would not be a problem if the RNG never generated 0. So, it must follow that it would also not be a problem if it never generated {1, 2, 3, ..., 253}.

That means that our RNG now only generates the values 254 and 255. Which of the values is generated is unpredictable on any given call. However, 7 of the 8 output bits are now always fixed and so completely predictable. Can you imagine how an attacker could exploit that?

Failing to generate only the number 0 is a weaker version of the same class of flaw.

“Pick all the numbers in the range with exactly the same probability” is a very important property of RNGs. Yes, skipping 16542 would be equally bad.

You can frame it around being “non predictable”, but then you need to define those words. It’s not, for example, a poker game where it’s trying to bluff you, right? It’s also not about just making predictions < 100% reliable and declaring victory. It must specifically make all predictions no better than random guessing, and that entails picking any number in range with equal probability, otherwise predictions like “it will be {hot spot}” or “it won’t be {cold spot}” do better than random chance. In this case, specifically, I can predict with 100% accuracy that the result won’t be 0, and that’s a flaw in its unpredictability. I can also predict a bunch of other things with slightly higher accuracy than random guessing, like that it will be odd or greater than max ÷ 2.

What are you talking about? The point is in fact to pick all the numbers in the range with exactly the same probability.

See section 7.3.17 of the Intel SDM, and how NIST SP800-90A (which the SDM refers to) defines "random number".

"Random" is used by most people to mean "random with an even distribution".

A weighted die is still random, but with an uneven distribution. This is effectively a 2^16-sided, weighted die.

Betty from accounting will have words.

That may well be a problem, yes.

“Predictable” and “not pick all numbers in range with exactly the same probability” are synonyms here.