That is a large concern of mine as well. I asked about it earler, but the question was treated as a joke, rather than seriously answered.
I'm sure I'm not the only Catholic on FR who has had to explain the difference between true papal infallibility and the false notion of papal impeccability to Protestants (and some atheists). It is frustrating to explain over and over that not every word from the Pope's mouth, nor every action he takes is considered infallible.
I have a hard time seeing where you draw the line between the true notion of a Council's infallibility - which relates specifically to the definitions and decrees issued by the councils - and the false notion of a Council's impeccability, which would presumably relate to everything else a Council says or does.
This is not to say that statments and teachings which are not infallible are not important. It is also not to say that a Council exercizes no authority other than infallibility. Both of those notions are clearly false. But to casually treat the infallibility of a Council is to invite the same error (i.e. equating infallibility and impeccability) which we so frequently see applied to the pope.
Jesus and Mary are Impeccable. All Ecumenical Councils are Infallible, it is intrinsic to their nature. With all due respect, it is you that is confused. Try to find any Catholic, traditionalist, modernist, conservative, that has ever described an Ecumenical Council as "impeccable."<>