There are lots of books on the development of the canon. For the most part, I just accept what has been handed down to me, but I do try to make some checks with what others have written or said.
I do not put Jesus, Paul, and Josephus into the “no evidence” category like you seem to do. I think they were doing their best to refer to some collection of inspired books and the best candidate by far that I can see for that collection is the Tanakh. And I expect to be judged by God for my assessment. I do think that the Tanakh and Apostolic Scriptures/books of the NT were/are inspired by God, but ultimately it is an act of faith. I have a well developed inner skeptic that is a part of my faith; however, in order to start anywhere, one has to act in faith based on what God has revealed to oneself.
And remember the NT writers didn’t quote from all of the books of the OT so we really don’t know what they considered to be Scripture. The LXX is a good candidate. And what do you do about the references to books that have been lost? The writers considered them to be Scripture but still the books are gone.
I think the Jesus, the authors of the NT books, and Josephus considered the Hebrew Tanakh to be the books of Scripture, but since they were writing in Greek they were often willing to take their quotes from the LXX, but this does NOT mean that they considered all the books in the LXX as Scripture. My point is that just because there is a reference to another book in Scripture does not mean that book is Scripture, it depends on how it is referred to. We know Paul referred to some parts of Greek philosophy contained in books, those are certainly not Scripture.
One was that Scripture is referred to is “It is written …” Another way is to refer to the author as understood by 1st century Jews, such as Moses or Isaiah.
There can be more oblique ways, and each assessed in turn. Here is an example: ESV Joh 10:22 At that time the Feast of Dedication took place at Jerusalem. It was winter,
ISV Joh 10:22 Now Hanukkah was taking place in Jerusalem. It was winter,
This is a reference to the non-Mosaic festival instituted in Esther. I do not think that this reference means that Esther must be considered Scripture, but it is evidence that the Jews at that time considered it authoritative enough to have the festival then and indeed they celebrate it today.
Jesus I trust. Josephus not so much. The NT authors were probably not looking up their quotes in a copy of the LXX. The fact they get so many of them slightly wrong suggests they were quoting from memory and in some cases actually changing the quote to fit their message. If they were willing to rewrite what was written how does that fit into your view of Scripture?
The fact that the Jews considered it authoritative made it Scripture for them. Correct me if I am wrong but you seem to be dancing around not taking some portions of the LXX as Scripture. If I might ask what do you have problems with?
Actual copies of any scroll of Scripture was rare as most were not literate. In an oral culture, the ability to read and especially write was a special skill, like being an auto mechanic today. If one needed something written in order to keep a semi-permanent record (like a marriage ketubah), one would go to a professional scribe to do it.
It was normal for a NT author to refer to Tanakh books via a recognizable reference. And going by memory, there might even be some looseness (that is, not being exact wording) in the reference. Any translation, including the LXX, involves interpretation and it seems at times the interpretation made by the LXX was preferred over just translating the Hebrew text itself into Greek. Given that the books of the NT come down to us in Greek and the LXX was already in Greek, I think it is significant that not all of the NT quotes are from the LXX. My take is that when the NT quotes the LXX as Scripture, the author/speaker of the NT book was endorsing the LXX’s interpretive translation of the Hebrew text. And then perhaps when they did not, then they were not endorsing it, at least that is my starting assumption.
I do not think the books of the LXX are inspired by God in the way that the Hebrew Tanakh was/is. That is, the LXX translation was a work by people who could make mistakes and lose things in translation, just as any translation can. It is important to know that the LXX exists and is quoted some 80-90% in the NT. But I only accept the LXX quotes in the NT as being approved translation interpretations of the Hebrew Tanakh. This implied approval certainly does not extend to the books outside the Hebrew Tanakh except as being useful for understanding the cultural context.
One way to look at it is in an analogy. In math, having different axiom sets means one can get different results in one’s proofs. In some abstract sense, one’s canon acts in a similar fashion. The Orthodox Jewish, Protestant/Messianic, Catholic, and Orthodox Christian books in the Bible canon are all different, so it is not too surprising that each group ends up with different conclusions at times. I am not Catholic, but I have read that they use some books in their Deuterocanon (Apocrypha) to justify some of their beliefs. It makes a difference.
It can also make a difference in how one understands parts of the NT. I do not think Paul accepted the LXX books beyond the Hebrew Tanakh as being Scripture in any form; but if one thinks he did, then one can end up reading some parts of Paul’s letters in a very different way than I do.
There is another aspect. There are Hebrew books that were split because the LXX added vowels and so the Greek text did not fit on just one scroll. But the splitting was in some cases done in a way that broke apart the natural literary structure of the single book. Also book collections in the Tanakh were broken apart and re-assembled, which I think again led to loss of context. When Jesus referred to the Law/Torah, Prophets, and Psalms I think he was referring to the Tanakh, but because the claimed Christian ordering is different a reader may not even know this is a possibility. For many reasons, I much prefer the 3 major book collections of the Tanakh to the claimed Christian 4 major book collections. I think they meant well, but destroyed meaning in the process.
Here are a few more reasons I decline to see the LXX as Scripture, altho it is very important in order to help understand the NT books.
RSV Sirach 25:24 From a woman sin had its beginning,
and because of her we all die.
I think this is a misunderstanding of the story in Gen 3. But if accepted as Scripture, it can seem to endorse that misunderstanding.
Another aspect is that Gen-Kings in the Tanakh form a literary structure called the Primary History of Israel. By adding Ruth into that, I think it confuses things so that literary connections inside the Primary History that exist are now made (more) opaque. Ruth was written later and did deliberately copy some of the wording used earlier, but that does not mean it should be inserted into an already existing structure.