Infallibility was given to the Church, excersiced throught he teachings of the apostles. And the Bishops are the successors to the apostles. Why does the teaching have to be so explicit in Scripture that it is there when they speak in union with the Pope? So your first problem seems more basic, you are working from a point of view of Sola Scriptura.
It can be logically derived from scripture:
Looking at Mathew 28 “And Jesus coming, spoke to them, saying: All power is given to me in heaven and in earth. 19 Going therefore, teach all nations: baptizing them in the name of the Father and of the Son and of the Holy Ghost. 20 Teaching them to observe all things whatsoever I have commanded you. And behold I am with you all days, even to the consummation of the world.” it is clear he is talking to the Apostles as a group. And it was promised to be there forever.
In John 14, we have another promis to the group "“But the Paraclete, the Holy Ghost, whom the Father will send in my name, he will teach you all things, and bring all things to your mind, whatsoever I shall have said to you” ".
But in Mathew 16: "That you are Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it. 19 And I will give to you the keys of the kingdom of heaven. And whatsoever you shall bind upon earth, it shall be bound also in heaven: and whatsoever you shall loose on earth, it shall be loosed also in heaven. 20 " It is clear he was talking to Peter specifically.
And we know that Christ intended His Church to be unified and that it is built on the Peter. So the sign of unity, is that the Bishops are in union with the Roman Bishop.
It is best if you read chapters 32-38 (actually paragraphs) of St Thomas Aquinas’s “Against the Errors of the Greeks”
dhspriory.org/thomas/ContraErrGraecorum.htm#b32 it will help a lot.
I would also recommend the Proof from Tradition of the Church’s infallibility in the old Encyclopedia:
newadvent.org/cathen/07790a.htm#II
And finally Proof from Scripture and Proof from Traidtion of Papal Infallibility on the same page from the Encyclopedia.
All good sources. Again, abandon the Sola Scriptura mentality you seem dead set on and then it will be easier to come to believe. If you do not believe in the Tradition of the Church, you are focused on the wrong issue; your problem is much more basic.