Why are bishops, priests, deacons and other teachers of the Catholic faith no longer required to take the Oath Against Modernism?
I do not have any particular insight into the mind of Pope Paul VI nor have I read anywhere his stated reasons. What I can say is that all who are to be ordained make a Profession of Faith and an Oath of Fidelity.
In the Profession of Faith the individual recites the Nicene Creed and then declares:
With firm faith, I also believe everything contained in the Word of God, whether written or handed down in Tradition, which the Church, either by a solemn judgement or by the ordinary and universal Magisterium, sets forth to be believed as divinely revealed.
I also firmly accept and hold each and everything definitively proposed by the Church regarding teaching on faith and morals.
Moreover, I adhere with religious submission of will and intellect to the teachings which either the Roman pontiff or the College of Bishops enunciate when they exercise their authentic Magisterium, even if they do not intend to proclaim these teachings by a definitive act.
And here is part of the Oath of Fidelity:
In fulfilling the charge entrusted to me in the name of the Church, I shall hold fast to the deposit of faith in its entirety; I shall faithfully hand it on and explain it, and I shall avoid any teachings contrary to it…
…With Christian obedience I shall follow what the Bishops, as authentic doctors and teachers of the faith, declare, or what they, as those who govern the Church, establish.
The Profession of Faith and Oath of Fidelity would seem to render the Oath against Modernism redundant. Modernism is contrary to the teachings of the Church and the deposit of faith, both of which the man to be ordained has sworn not to contradict.