"She reads me a newspaper."
Translation:Αυτή μου διαβάζει μία εφημερίδα.
You're not wrong.
But while Ancient Greek had a dative case, it got lost on the way to Modern Greek.
DIfferent dialects differed in how they replaced it; in the north, it tended to be replaced by the accusative while in the south it was the genitive. This southern version was then used as a basis for the standard language, which is why standard Greek uses the genitive pronouns for indirect objects.
In Thessaloniki, you might well hear με in such a sentence, but it's non-standard. (I remember hearing a story of a woman on a bus asking the bus driver to Άνοιξέ με από πίσω, which in northern dialect means "open to-me at the back" = open the back doors for me, while to someone who learned standard Greek it can only mean "open me at the back" = open me up from behind!)
So με is wrong because this is an indirect object - the direct object is μια εφημερίδα and the "to me, for me" is an object that would be in the dative case in languages such as Ancient Greek or German, but in Modern Greek is in the genitive if you use the short unstressed form of the pronoun. (If you use the full form, it will generally be with preposition σε + accusative, e.g. σ' εμένα, στον κύριο, etc.)
There is no genitive for pronouns. There is accusative and dative. When you use the dative after a noun, the whole construction gains the meaning of "my...", but that's just it.
If you reverse the English sentence, you will see that there is a hidden 'to': She reads a newspaper to me.
That's why the dative is necessary here.
I've usually seen modern Greek described as having no dative case. Then μου is formally genitive, though it is used in standard Greek for indirect objects, which often take the dative in languages that have it.
The dative would have been something like μοι, I think, back when the language still had a dative.