According to a Dr. William Mahrt:
HERE IS NO CONCRETE HISTORICAL EVIDENCE for a third lesson in the Roman Rite. There is, apparently, in the Milanese Rite. Perhaps the evidence that has been relied upon was the very disjunction between gradual and alleluia. Evidence against the proposed historical order (and thus the present usage of the ordinary form) is that the assignment of alleluias to the Sundays after Pentecost varies from place to place, while the other propers are quite consistent from place to place; the alleluias were unquestionably assigned after the time when a hypothetical third (unproven) reading was the case. But for the Latin Rite, there is evidence that in Augustine’s practice, there was only one lesson before the Gospel, because he preaches on the lesson, the psalm (responsorial psalm), and Gospel.
He goes on to state further:
S REGARDS THE OLD TESTAMENT, we are repeatedly assured that there was an Old Testament reading each Sunday morning at Mass, but that quite mysteriously these all vanished by the seventh century, and vanished leaving no memory that they had ever existed: no homilies on them by Leo or Gregory, no inadvertent cross references to them in any surviving source, not one palimpsest listing one pericope and the Sunday to which it was assigned, no tradition as to what Pope suppressed them or why; just an a priori assertion that there is a reading missing between the Gradual and the Alleluia, which would, incidentally, place the Old Testament reading after the New, contrary to practice elsewhere in the traditional Missal. This argument from silence is wildly improbable.
There are indeed Old Testament lessons on penitential days in the traditional Roman lectionary, but these are quite a different matter. The alleged set of vanished Old Testament readings are, I fear, a romantic fantasy like the vanished people’s offertory procession. They are only a theory on the lips of a liturgist, like the smile on the face of the Cheshire cat that isn’t really there. If it is now thought desirable to introduce Old Testament readings, let a new three year cycle of them be drawn up and introduced, but on an optional basis, and not on the specious ground that some element due in the liturgy had disappeared. (source)
I remain somewhat convoluted in my thinking here. I was always taught that they were there, disappeared, and were restored with the great reform of the lectionary following Vatican II. Whether this is mythology or not, it is impossible to believe that they were not present in the earliest church. Indeed, there were Christians before there was a New Testament. The whole nature of the Christian faith from the Gospels to Acts is that the writings of Moses and the prophets all testified of Jesus. This is Jesus' own insistence to His disciples. It is incredible to believe then that the early Christians did not read the Law and the Prophets (as well as the poetic and wisdom writings) and not read Jesus Christ from them. That said, when the Roman lectionary was formed, it is entirely possible that the Old Testament readings were not including for the Mass propers but were relegated to the Daily Offices. Whatever the history, and it is far from settled at this point, the Christians today have benefited from a full restoration of the Old Testament readings into both the three year lectionary and the one year lectionary and for this we should all be glad.