This is getting off topic, but your questions are surely reasonable so I will try to offer a response. That said, I probably won't attend to this tangent for too long, since it will lead down all sorts of rabbit holes.
So how, outside of direct revelation, is one supposed to discern what Christ, or the Apostles hold without consulting the scriptures? Outside of what we can read in the scriptures (and direct revelation which I, at least, have not experienced), they are not available for consultation on which doctrines are correct and which are poppycock. [...] Additionally, the compilation of the scriptures was the work of the Church when those successors you mentioned were in agreement.
Yes, but it is notable that different Christians hold to different canons, and that the canon is obviously not included in Scripture. Also, we have evidence of disagreements among the early community even within the NT texts themselves, and history shows us that this did not disappear.
The Church does not seem to have unanimity of doctrine throughout its many denominations so how can a divided Church have authoritative advice? Additionally, the compilation of the scriptures was the work of the Church when those successors you mentioned were in agreement. Shouldn't we take advantage of that in deciding the matter of what doctrine may or may not conform to Christ?
Yes, I think so, and I think most Christians do. For example, in general Christians take the first seven ecumenical councils to be more authoritative than anything that came afterwards. This seems quite reasonable to me, and is one way to address the fact that Christianity is divided.
Certainly, we can pray for enlightenment from the Holy Spirit but trusting ourselves to be honest with ourselves about what the Holy Spirit may be telling us seems more likely to be a problem than trusting the Holy Spirit to guide us by using the scriptures. One can claim the Holy Spirit tells us something the scriptures do not, but that claim is not as solid as being able to point out the Holy Spirit tells us the same thing that scripture does.
The Scriptures do have authority, but it doesn't ultimately make sense to say that they constitute the sole rule of faith ("
Sola Scriptura").
The authority of the Bible is not the authority of a book but the authority of God's Word. It is a written account of what God has told his people about himself from the time of Adam to the time of the early Church.
If it stands apart from any living voice which interprets and guides, then it is 'a book'. The Protestant's ultimate authority is an isolated book, albeit inspired. In Catholicism (and elsewhere) this book is part of a tradition, and flows from the
living voice of God which continues to speak today, to interpret and guide. These hard logical contradictions noted in the OP do not apply to the Apostolic paradigm. The book needs to be seen in light of a greater whole.
As for the point. You claim that Sola Scriptura is self-refuting and therefore not useful.
Yes, but be careful not to conflate
Sola Scriptura and Scripture itself.
Sola Scriptura is a Protestant doctrine which is very different from Scripture itself. Scripture is useful and indispensable;
Sola Scriptura is not. Protestants are blessed with the knowledge that the Scriptures are God's inspired word, and they should in no way forfeit such a divine gift.
The point is self - refuting in that it is a said to exist without having one quality of anything that does actually exist in the physical world.
A mathematical point is not a physical object, and thus should not be expected to exist in the way that objects in the physical world exist.
That being the case, if both are self-refuting, and being self-refuting automatically makes the one useless, how does one explain how the other is not only useful but necessary?
I don't see how a mathematical point can said to be "self-refuting." Does it entail some proposition which falsifies its own truth? I think not. A mathematical point is a useful abstraction that is used for the sake of other (mathematical) abstractions. It does not exist physically, nor does it's definition entail that it itself is false. Indeed, a mathematical point is a (mental) object, not a proposition, so it can't entail or refute anything.
I think your larger question is as follows: "Maybe
Sola Scriptura isn't ultimately defensible, but are the alternatives any better? Are there better sources of Christian authority than the
Sola Scriptura model?" Now I grant that
Sola Scriptura seems to become more plausible in light of the divisions of Christianity, but I do think there are better models. In general I would say that although the various options among the Apostolic Paradigm are not perfect, they are nevertheless not
self-refuting. Self-refutation is a rather serious problem. I realize I have posed you with a problem and no good solution, but I would suggest that any of the options which adhere to the Apostolic Paradigm are better options than
Sola Scriptura, whether that be Catholicism, Eastern Orthodoxy, Oriental Orthodoxy, etc.
Finally, in keeping with the strict spirit of the OP, I should again stress that
the Protestant can abandon Sola Scriptura without abandoning the inspired Scriptures. Doing so only means that you will be open to God's guiding and interpretive word outside of the Scriptures. If you encounter someone who claims to represent that divine guidance, but contradicts the Scriptures, then you know he is not of God (cf. Gal 1:8). At the same time, you should be open to the idea that God has not left his Church without guidance, and that that guidance will look somehow similar to the guidance he provided in the Apostolic Age.