Not everything taught in an Ecumenical Council is necessarily "dogma". Only things of faith and morals. Not discipline. For example, a council teaching it is incorrect to kneel during the Mass is not an example of dogmatic, unchangeable teaching, a teaching of revelation given by Christ to the Apostles. This would be "tradition", small-"t", one that is a matter of discipline. Another example would be whether priests can marry. Orthodox allow it, Latins normally do not, although they make exceptions. Again, one asks "is this a matter of faith or morals for all time?" If yes, then we have an infallible statement - when in union with the Bishop of Rome. In Honorius I case, I don't think we have a matter of faith and morals. In any case, Honorius's problem was not faulty teachings, but not pursuing heresy sufficiently when he saw it.
Remember Peter denied Christ - yet he became the leader of the Apostolic community. What is important to remember is that God guides His Church through a special charism, a gift of the Spirit, but only regarding teaching faith and morals. It is fairly limited. Why does God allow error to "creep in"? Why does God allow sin? I think one would be hard pressed to say that the Gospel has been lost. The Church is doing its job, often times, in spite of people within the Church!
Regards
And a sad footnote about Peter. He failed in his mission.
Galatians 2:9 And when James, Cephas, and John, who seemed to be pillars, perceived the grace that was given unto me, they gave to me and Barnabas the right hands of fellowship; that we should go unto the heathen, and they unto the circumcision.
If anyone could lay claim to being "pope" it would be Paul, not Peter.
I was specifically asking about the actual beliefs of Honorius I. It appears that is the minimum he was held accountable for. Since an infallible Council declared that he had heretical beliefs, is this not a matter of faith and morals? How can a heretic be whole with the faith?