First, speech is different, because of intonation; it is because we don't have intonation in writing, that we have punctuation to create the boundaries. I mean that intonation itself can act as the quotation marks, in regards to speech.

At the core, the idea is that the quotes are what lift the word out of the case system, or at least, tell the reader that they are momentarily off the railroad track of the rest of the sentence.

Without the quotes, I'd propose, the sentence ought/need/should be:

"Её зовут Катей."

Otherwise you just have another Nominative Animate Person floating around, and the sentence wouldn't be consistent with the systems holding up the rest of the language.
It's essentially because it keeps an important consistency with the rest of the language.

*"Они считают меня глупый."

For instance, I believe, very few steps from the original phrase is the phrase:
"Её зовёт Катя" or even "Её зовут Катя и Катюша"
As far as I can tell, the use of this "Determinant Instrumental" in absolutely necessary here. Otherwise, Катя could be doing something or some person could be calling some person Катя. And the second sentence is even worse!

I think the only reason it's not completely standard to introduce people this way is because you'd be introducing a name in its "non primary" form, so to speak; and for some reason that is naturally unsettling.
It's almost as if the case could be made for Катя to be considered "vocative case" in these expressions.

Now I don't fully expect every person to make sure to add quotation marks; the expression is practically a set phrase that only has this problem with introducing names; and all this grammar upkeep only really needs protection for the sake of the people who care enough to need to use it (academic writing). You can always just use poor grammar and then leave meaning to context (essentially: just hope real hard the reader can read your mind as well as they can your words) but coming from an English speaker's perspective it's not exactly a fun experience.

Basically, all it takes for the "великий, могучий, правдивый и свободный русский язык" to become none of that is a single crack in the hull.
And having a language so powerful, I think, specifically lessens the distance that the average person is from being able to speak in effective, productive speech; the less powerful - the bigger a hole one has to learn themselves out of.