While I disagree that the Anglicans had in mind the Oriental non-chalcedonians (why would Jerusalem be listed), it seems more likely its over the filioque, and our Orthodox belief concerning ecclesiology.
The Anglicans listing the Athanasian creed is problematic. Not only for its usage of the fillioque but that it is a poor expression and possibly even a cacodox expression of the Trinity. If i remember correctly it also avoids refering to the Virgin Mary as Theotokos. Its a western post Chalcedonian Creed which relegates the first three councils to the back seat. The overemphasis of Chalcedon while burying Ephesus has had repercussions within protestantism.
My reading of the article was an attempt to account for the presence of Antioch, Jerusalem (which, on my reading, would have been deemed "Nestorian" by those who drew up the articles) and Alexandria (which, on my reading, the framers would have deemed "Monophysite") and the absence of Constantinople.
My reading is non-traditional, however, so I won't be hurt if you reject it. I'm not sure of it myself. The traditional reading of the Article is this: The churches of Jerusalem, Alexandria, and Antioch, as well as that of Rome, all claim apostolic connections: Peter, along with James the Lord's brother, were leaders of the Jerusalem church. Antioch also claims a connection to Peter. Alexandria claimed to have received the Gospel from Mark, a disciple of Peter. Yet all these churches, at one time or another, had heterodox bishops: Peter the Fuller in Antioch, Dioscoros in Alexandria, Sallustius in Jerusalem. These churches who received the Gospel from the Apostles were not thereby protected from being ruled by erring hierarchs. Hence it does not follow from the Roman church's traditional association with Peter that its bishops will always be free from error, either. (Papal infallibility was not officially proclaimed until Vatican I, but it was claimed well before then.) On this traditional reading, Constantinople is left out because it did not claim to have received the Gospel from Peter or anyone in Peter's generation. Hence the three cities mentioned were produced by the Article's framers as historical evidence against Roman claims.
In 1869 Ecumenical Patriarch Gregory VI sent a letter to the Archibald Campbell, Archbishop of Canterbury, which included the following:
On descending to the particulars of the contents of the [English] Prayer Book, and of the distinguished Confession of the Thirty-nine Articles attached to is,--since, in the perusal of them, both the statements concerning the eternal existence of the Holy Spirit...and those concerning the Divine Eucharist, and, further, those concerning the number of the sacraments, concerning Apostolical and eccleiastical tradition, the authority of the truly genuine Oecumenical Councils, the position and mutual relations of the Church on earth and that in heaven, and, moreover, the honor and reverence due from us to those who are the contemplative and active heroes of the faith--the adamantine martyra and ascetics--since, we say, these statements appeared to us to savor too much of novelty, and since that which is said on page 592 (Art. XIX.), 'As the churches of Jerusalem, Alexandria, and Antioch have erred, so also the Church of Rome hath erred, not only in their living and manner of ceremonies, but also in matters of faith,' depriving, as it does, the Eastern Churches of soundness and perfection of faith...throws us into suspense, so that we doubt what we are to judge of the rule of Anglican Orthodoxy; we will, therefore, pray with our whole soul to the Author and Finisher of our salvation to enlighten the understanding of all with the light of His knowledge, and to make of all nations one speech, of the one faith, and of the one love and of the one hope of the Gospel...
The English Church's Convocation recommended that article XIX be explained to the Patriarch with the traditional explanation I have given above: that the point was to deny that a bishopric's ancient Apostolic connection could guarantee inerrancy on the part of the see's bishops.
So, either by my new-fangled reading and the traditional one, Article XIX has nothing to do with the filioque. The filioque is slowly being phased out in Anglicanism anyhow.
The English version of Article VIII does, or once did, list the psalm Quicunque vult as a creed that should be accepted by believers because it can be proved from scripture. You should be aware, though that the American version of Article VIII lists only the Nicene Creed and "that which is commonly called the Apostles' creed". The American Episcopal Church has never used the Quicunque vult in public worship.