A pronoun is an interesting animal, isn't it!
It certainly makes one uneasy as regards one of the tests of functional words: like a noun, it seems to supply content and can serve as a head of a phrase. In fact, it can do most things a noun can do.
One intuitive resolution to that idea that it serves its own content, though, might be to compare it with variables in programming. A variable is generally not a literal, i.e. it isn't itself a value but a rule for how to look up a value. However, you can substitute it anywhere that you can use a value:
>>> multiply(5, 3)
>>> x = 3
>>> multiply(5, x)
x has no meaning "on its own". If we hadn't supplied a referent for
x, the second-last line would cause an error since it's a language-level operation requiring existing content. That's a pronoun.
Coming back to linguistics, another test is whether the class is open or closed. Can we add to the class of pronouns? Can we derive new words from them?
The canonical answer is no: there's a finite set of roles distinguished in English and we already have a pronoun for each of them. We also have a fixed correspondence between pronouns and derived forms (he : him :: she : her, etc.), and this relationship is immutable.
But here, too, pronouns do display some noun-like qualities. For example, some members of my family say "youse" to distinguish second person singular and plural. For consistency we'll say that's a nonstandard dialect. Then if I borrow their usage, am I code-switching or have I expanded the set of pronouns in my dialect? It sure seems easier to do than to create a new conjunction...
"Okay, but that's because there's a lacuna in English's paradigm. It's still a finite set." This brings us to the modern debate! If pronouns distinguish by gender, and we define the set of genders as non-finite, don't pronouns become an open class? Suddenly linguistics is politicized. Do we revise our categorization of pronouns, or do we say that words like xhe, xe, xir are really nouns? Could the attempt to use them as pronouns account for some of the resistance (without their actually knowing why it's a hurdle to use them naturally)?
In short, pronouns do in one sense fulfill a functional rather than a content role, as seen above, but they exhibit strange behaviour for function words.
I raise these questions without knowing how they are resolved by those who study this area specifically, and would be interested to read more in another answer or comment!