I can well understand the term "Crossing the Rubicon" as denoting taking an irrevocable step, as Caesar did when he came home to seize power.
I can understand "Crossing the Delaware" as a significant and symbolic step in reversing the fortunes of war, as Washington did in the unpleasantness between the Crown and its American colonies.
And "Crossing the Rhine" was what happened in the Volkerwanderung of the German tribes into the Roman provinces and was a significant step in the Allied conquest of Germany in WWII.
We can come up with lots of allusions to crossing rivers.
But the only context I had for the Tiber was that it was a bridge on that river that Horatius defended in his famous stand against the Etruscan invasion of Rome, celebrated in the epic poem "Horatius at the Bridge" by Thomas Babington Macaulay.
Perhaps it is simply that the Tiber is no longer a river on the approaches to Rome but instead flows through Rome and divides the Vatican from Rome's "city centre". So "crossing the Tiber" is simply a symbolic description of the process of returning to Holy Mother Church.
But I don't think that one "crosses the Tiber" by becoming more educated about the Church. Religious conversion, in any direction, is not an intellectual exercise; albeit the intellectual exercise may, for many, have been a condition precedent for a true conversion.
Well, actual education may tend to remove a few preconceived notions. And those preconceived notions may make it far more difficult to perceive and respond to the calling of the Holy Spirit (again, an IMHO)