I'm sure Newman or someone has already made this argument, but here it goes:
It is asserted that the Church of Jesus Christ is fundamentally invisible. But this cannot be. The Law of Non-Contradiction states that a thing cannot be and not be in the same way at the same time. But this "Church" proposes as de fide doctrines which are contradictory, such as in the doctrine of the Eucharist. As it is written, "A house divided against itself cannot stand." But the Church is the "pillar and foundation of the truth." It is also called the "household of God." God cannot be divided, as it is written, "Hear O Israel, the Lord your God, the Lord is one." Therefore, this concept called by the same name cannot be the Church Christ founded.
Corollary: Now it is clear that a man may dissent from the true faith. But it does not follow that merely asserting a different set of propositions makes those propositions true. Based upon what we have already stated, the organs by which the contradictions come to be known (often called "churches" themselves) cannot be the Church, unless one set of propositions be true, and the other false. But that is not admitted by the concept. Nor can the communities be smaller parts of the same whole, because they do not agree upon that which is de fide. Now, the formation of those communities, with respect to an assertion of legitimacy, depends upon the concept which has proven false, as was shown above. Therefore, it is not reasonable to assume that the content of those propositions is true, because they are the fruit of a false concept. Therefore, the Church of Jesus Christ must be visible. Moreover, it is a likely supposition that one who holds propositions derived thence (that is, from the false concept) is in dissent (heresy) and schism.
It is asserted that the Church of Jesus Christ is fundamentally invisible. But this cannot be. The Law of Non-Contradiction states that a thing cannot be and not be in the same way at the same time. But this "Church" proposes as de fide doctrines which are contradictory, such as in the doctrine of the Eucharist. As it is written, "A house divided against itself cannot stand." But the Church is the "pillar and foundation of the truth." It is also called the "household of God." God cannot be divided, as it is written, "Hear O Israel, the Lord your God, the Lord is one." Therefore, this concept called by the same name cannot be the Church Christ founded.
Corollary: Now it is clear that a man may dissent from the true faith. But it does not follow that merely asserting a different set of propositions makes those propositions true. Based upon what we have already stated, the organs by which the contradictions come to be known (often called "churches" themselves) cannot be the Church, unless one set of propositions be true, and the other false. But that is not admitted by the concept. Nor can the communities be smaller parts of the same whole, because they do not agree upon that which is de fide. Now, the formation of those communities, with respect to an assertion of legitimacy, depends upon the concept which has proven false, as was shown above. Therefore, it is not reasonable to assume that the content of those propositions is true, because they are the fruit of a false concept. Therefore, the Church of Jesus Christ must be visible. Moreover, it is a likely supposition that one who holds propositions derived thence (that is, from the false concept) is in dissent (heresy) and schism.
Comments
Further, in the time of Christ, we see that the "Church" was divided into divisions such as the Pharisees and that the powerful within those divisions were in error against God. Therefore, if anything the Old Testament declarations argue for an invisible household -- those who embrace the Covenant with the Heart -- as an inner core within the visible nation of Israel and against the idea that the visible entity is the entity of greatest concern.
Finally, you do not leave room for the range of usage in the nature of language. We must remember that language is not always used the same way in Scripture. So, while at times Israel may refer to the nation, other times, it clearly is oriented toward the Faithful Remnant. At times, "Sons of Abraham" refer to the descendants of Abraham, yet Christ himself says God could raise up "Sons of Abraham" at will.
This is even so on the nature of God, whom we would agree with the Angelic Doctor concerning; God is "simple" being composed on one, indivisible unique Godhead. Nevertheless, we without issue avoid the heresies of Eutyches and Patripassianism when we recognize the divisions within the Trinity and reject overzealous applications of the communication of Divine Attributes between the persons of the Trinity.
You didn't even attempt to answer the question: How can the same Church of God propose contrary doctrines for belief de fide? It creates two unacceptable alternatives: 1) God lies (impossible), or 2) one system is correct and others are not (which the ecclesiology doesn't permit). Why don't you stop worrying about the Catholic implications I might draw, and answer the question? It's not a visibility question per se; it's a dogmatic one.
And, of course, the Pharisees were amongst the closest. But "closest" doesn't count when you are whitewashed tombs that obey the law and have the institutional support but fail to be within God's covenant with the heart -- when you reject the very fulfillment of that covenant even.