Both they should unify these things or bring a little closer in uniformity. I can't believe the Apostles did things so differently from one another and yet they are where we receive our ecclesiastical traditions.
It is a pious but ultimately fanciful and naive notion that the traditions (lower-case t) come directly from the Apostles in any substantial sense, especially in any particular detail.
In fact, the Introit itself is known to have been invented at a certain date centuries later, though likely developed from the singing of psalms in the synagogue.
The Catholic Encyclopedia (1917), says:
The "Liber Pontificalis" ascribes this antiphonal chant at the Introit to Pope Celestine I (422-32): "He ordered that the psalms of David be sung antiphonally [antiphonatim, by two choirs alternately] by all before the Sacrifice, which was not done before; but only the epistle of St. Paul was read and the holy Gospel" (ed. Duchesne, I, Paris, 1886, 230). The text seems even to attribute the use of the Introit-psalm in any form to this pope. Medieval writers take this idea from the "Liber Pontificalis", e.g. Honorius of Autun, "Gemma animæ" (in P. L., CLXXII): "Pope Celestine ordered psalms to be sung at the entrance (ad introitum) of the Mass. Pope Gregory afterwards composed antiphons in modulation for the entrance of the Mass" (I, lxxxvii). Probst thought that Gelasius I (492-96) invented the Introit (Die abendländische Messe vom 5 bis zum 8 Jahrhundert, Münster, 1896, 36).http://www.newadvent.org/cathen/08081a.htm
We don't have to do exactly the same thing but this is odd and radical difference.
"odd" and "radical"?!?! lol. It is an extremely minor difference considering how different the liturgies as a whole are.
In fact, a bigger difference is certainly that the East doesnt have an Introit at all!
Again, Catholic Encyclopedia says, in its article on the Introit:
There is nothing corresponding to our Introit in the Eastern rites. In all of them the liturgy begins quite differently. The preparation (vesting, preparation of the offerings) takes place in the sanctuary, so there is no procession to the altar.
Either the East should stop the laity or the West should have the laity sing the Propers.
No, there is no need for this sort of monolithism of tradition. There is a diversity of Rites and a diversity of traditions in the Church
Somewhere and at sometime in Church history we both must have done the Mass the same general way with our little particular differences from culture.
Not really. The differences between different liturgies are hardly "little" and except for perhaps the first few decades when the apostles were still alive...they have always been said very differently.
I want to know what the Apostles has their faithful do and that it what we ought to do.
There was no concept of "Propers" in that sense for the Apostles, certainly not the "Introit" in particular, and no choir of clerics yet existed. Only the distinction between priest and laity at the beginning.
Also, no there is no need to take archaeologism as the paradigm.