No, it shouldn't.
"my newspaper" would be η εφημερίδα μου (or in the accusative, as the object of "she reads...", την εφημερίδα μου) -- the genitive short-form pronoun such as μου comes immediately after the noun and there is usually a definite article before it.
Αυτή με διαβάζει would mean that she reads me (i.e. she is using me as reading material; perhaps I am a book?).
Αυτή μου διαβάζει is "she reads to me" (i.e. I am the audience) -- using the genitive pronoun as a replacement for the dative case which was lost.
English just has one "object" case for pronouns which is used for both direct object (e.g. "she reads me = I am a book") and indirect object (e.g. "she reads me a story = I am a listener"). But standard Greek distinguishes those two uses.
In the north, you might well hear Αυτή με διαβάζει for "she reads (to) me"... but that's not standard.
In this situation when an indirect object is needed,the pronoun -μου- in its genitive case is indeed more appropriate. But, when it is used in the place of a direct object the accusative form -με- should be used instead. So standard phrases like "Αυτός μου λέει" (he tells me) should be Αυτός με λέει rather. I know that this is not the standard type and it is used especially in northern Greece but this is more correct.
Your example is not a direct object, but an indirect one, still (Αυτός λέει σε εμένα=μου λέει and not αυτός λέει εμένα=με λέει). In Northern Greece, accusative is used for indirect objects when a direct is not present. Με έδωσες can mean both "you gave me (to someone)" or "you gave to me". So, in standard Greek one always uses the genitive case of pronouns for indirect objects and always the accusative case for direct objects. In Northern Greece, this rule breaks.
I get what you mean but I think that the standard rule is to put the object in the accusative case if it receives the verb's act. For example σε ταΐζω το φαγητό σου (I feed you your food) has its indirect object in accusative as well. Moreover when using plural persons with the verb λέω all Greeks tend to say "τους λέω"(i tell them). If a genitive case was necessary this type would be "των λέω" which sounds highly wrong. So the standard type of greek uses different cases in singular and plural while the "northern" type is more accurate. This has to do with the fact that when the dative case was abolished and modern greek took its final form some verbs whith objects in dative case had to replace it. (Λέγω "τινί" τι). And the closest to this usage of dative case is the accusative form.
First of all, ταΐζω εσένα and not ταΐζω σε εσένα, so σε ταΐζω (σε is direct object, ταΐζω gets two direct objects ταΐζω κάποιον κάτι as opposed to λέω σε κάποιον κάτι). Second, αυτοί is declined this way:
Strong form/Weak form
αυτών/τους (των is used after some archaic prepositions only)
That's why we can say for example, Τους τους έδωσα=I gave them to them. (what you say actually happens because the weak form of the accusative and genitive case of plural pronouns is the same, in εμείς and εσείς not only the weak form, but also the strong ones), see here Old type genitive εμού, εσού, ημών, υμών,των as weak genitive form etc are used rarely after archaic prepositions and nowhere else. When the dative case was abolished, it was replaced with preposition (σε) + accusative and with the genitive in pronouns in southern Greece. So, there is not really a different cases usage, at least not more than in Northern Greece (it is just that official Greek uses genitive pronouns instead of preposition + accusative and northern greek idiom uses accusative instead of preposition+accusative, which sounds less "correct" to me, as it can be mistaken for a direct object and not an indirect one). This happened for reasons analyzed here
Replacing an object case with a prepositional phrase, especially when another existing case describes the same meaning does not follow the rule from a grammatical perspective. But for sure in a language like greek with so many local linguistic diversities we could find tons of such examples. What I wanted to mention from a scholar point of view was that something is not necessarily wrong if it is not the standard