"We are infallible"
"How do you know that?"
"We say so, and we are infallible."
Doesn't meet the sniff test. And good lord, extrapolating the "gates of hell shall not prevail" language into infallibility doesn't cut it.
What is believable is that so many spiritual people have prayed and debated this stuff for centuries. The Holy Spirit guided them and inspired them, but did not ask them to take dictation...