No, it would not be justified.
If papal infallibility was not true and well-established, then defining infallibility ad-hoc in 1870 would have been mortally dangerous for the Church. At that point, there had been 255 popes over the course of about 1800 years. Note that ex-cathedra is not a label; it is basically a clear statement on a particular topic, made by a particular person, and directed at a particular group of people. That's all. If it had turned out that any of these 255 popes had made conflicting ex-cathedra statements, then the doctrine of infallibility and seat of Saint Peter would have been undermined. It does not take much to speak ex-cathedra, and though it is not an everyday occurrence, many clear ex-cathedra statements have been made. Why would the Church bet the house on papal infallibility in the (rather plain and sensible) form of ex-cathedra? Why not rely on other organs such as Ecumenical Councils or some form of consensus among bishops in order to get infallibility?
Because the outcome was known in advance. The Church was certain that no two popes would be found to make conflicting ex-cathedra statements, because it was known all along and for a very long time that the pope was indeed to be considered infallible in this way.
It's actually too bad that it was not defined "ad-hoc" in 1870. That would constitute a clean scientific test of infallibility. The prediction would have been: no two papal statements of a particular form shall be found to contradict each other. Not a trivial prediction! Oh well. I think I prefer long-standing tradition to a confirmed scientific test.
The section Proof of the Church's infallibility in the Catholic Encyclopedia, the source of the quote in your question, lists much evidence and many proofs for papal infallibility. I will not repeat them, since that would make the answer tedious. I think the above is a much cleaner argument.