What is the logic behind that? Teaching I can see as that is what Paul commanded. But reading Scripture in a service isn't allowed by anyone but the above?
A priest must preside at the Mass, because only a consecrated priest is enabled, through having received the sacrament of priesthood, to consecrate the bread and the wine—with Christ acting through him.
Since the priest presides at the Mass, it is a reasonable rule that he should read the Gospel, while a lay person or persons are often or usually chosen to read the epistles.
That is at Mass. Of course, any Catholic—or anyone else—is welcome to read the Gospels, either to themselves or aloud, outside of the Mass. It would be perfectly legitimate, for example, for a lay person to read the gospel to others at a Bible study class.
This rule is not necessary, per se, but it is the rule, as set down by the Church, because the Gospel selections, in which Jesus himself usually appears and speaks, are honored above the two other biblical texts read at every Mass.
If there is a deacon present, then he, too, is permitted to read the gospel.
These are not unchangeable laws, but they are reasonable ones, as set down by the Church and by long custom.