We know that the Quran was complete in the last year of the prophet's life and was put in a book (Mus-haf) since then, so there was no opportunity before then for the prophet (pbuh) to swear on the Quran. And just because the prophet (pbuh) didn't do something doesn't mean it cannot or should not be done. The only things that should not be done are the things God or His messenger explicitly forbade, or other evidence to forbid them can be clearly shown.
Having the Mus-haf there during swearing does have a deterring effect on some people! There are people who may give false testimony but when they have to put their hand on the Mus-haf as they swear, they may think twice before lying
That said, making this a regular practice borders on Bid`a (innovation in religion) and thus is undesirable. The fact that this is also an imitation of the Christians who swear on the Bible, adds to the argument that this practice is not a good one if it becomes a habit or becomes systemic. That's because the prophet (pbuh) often forbade Muslims from imitating non-Muslims.
Finally, the Sunna in swearing was given by the prophet (pbuh) when he said, "Whoever is swearing, let him swear by God or be silent."
God knows best.