What's behind the word "size issues"?

Yes, if you have global choice, then you can prove the principle you speak about, which is known as Limitation of Size.

However, in most cases where the informal "too big to be a set" argument is used, one is generally assumed to be working in ZF(C), where neither global choice nor limitation of size can even be stated. Generally, what this informal argument means is something like

If the collection of all the things you're speaking about there constituted a set, then it follows that there would also need to be a set of all sets (or a set of all ordinals), in which case Russell's paradox (or Burali-Forti's) would obtain a contradiction. So your collection definitely does not specify a set.

Here, the step where you conclude "there would need to be a set of all sets" often involves Replacement, but not necessarily. For example, the collection of all singletons is "too big" just by virtue of the Axiom of Union -- and similarly, taking the union a small finite number of times will be enough to show that collections like "all groups" or "all small categories" are too big to be sets.

The point is that Replacement guarantees that a class which can be put in bijection with a set, is a set. Here class means a definable collection, and a bijection means definable bijection.

If every set has a cardinality, then proper classes are "too big" to have a cardinality. Which is exactly the idea.

The thing is, that assuming global choice, every two proper classes are equipotent. So there is just "one size" of proper classes.

Without Replacement it is possible that there is a proper class which is countable, in the sense that there is a definable bijection between the natural numbers and the class. And if that is not a motivation to accept Replacement, I don't know what is...