There’s no very good reason to read the books of the Bible in order. The order (and even the selection of which books are canonical) is an artificial construct devised long after the individual books were written.
Most of the Old Testament consists of Jewish history, beginning with creation. There are a few that could probably stand on their own: Job, Jonah, the Song of Solomon, Proverbs, and maybe Psalms. (You would miss a lot in Psalms doing this.)
The New Testament is full of allusions to the old and cannot be understood correctly without reading it first. It probably doesn’t matter the order you read the first four books, or even if you read all four - Matthew, Mark, and Luke are very similar in tome and they describe the same events, but you wouldn’t want to skip John.
Acts is simply what happens after the four gospels, and the remainder of the Bible consists of epistles from the apostles which refer to events in Acts and in the gospels, and it wraps up with Revelations.
I don’t really see how you could read it in any substantially different order and get as much out of it.
A few examples: John, the fourth book of the New Testament, was one of the last written.
Luke and Acts are probably part of the same document, written by the same author, but are separated in the traditional ordering.
And so on...
Edit (Correction): as the person replying to me correctly points out, Luke and Acts were not part of the same document, though they were probably written by the same person.
Luke and Acts were probably not part of the same document, but were written separately. The beginning of Acts supports this when the author speaks of his previous book and re-addresses his audience. I suppose you could argue it’s all an interpolation, but the books were separate before the canon was established.
Either way, I wouldn’t consider it arbitrary to have all the gospel narratives together and then have subsequent events afterward. Perhaps you could argue that Luke and John should be swapped, so as to proceed from Luke to Acts, but then Luke is much more like Matthew and Mark theologically and stylistically. You could set John first, but it’s the most difficult of the four. It’s hard to see how your example is a good one of arbitrary ordering. Four books covering the same material are all in one place; the three synoptic gospels are in the same place - how else would you do it?
Moreover, the idea of reading New Testament without having read the Old one sounds rather ridiculous if you aim for anything more than assorted crumblings of the holy words. People at the time of Jesus were living under the law of the old books—it defined their thinking. How can you have an understanding any similar to the intended one if you circumvent this inherent order and go for walls before the foundation? It's like watching Spaceballs without having a glimpse of Star Wars.
So read the most relevant OT books and get on with it! Genesis, Exodus, Duteronomoy, Ruth, a little of the major prophets, Job, Proverbs 8, Psalm 22,23,91 I think? I'm probably missing a lot but seriously it's not needed if you want to understand NT context. I mean the rabbinical culture that Jesus was dealing with in the Gospels was based on a side-conversation the Jews were having about the Torah, not even about the Torah itself. So you'd have to read the Mishnah as well if you really wanted the full context.
I think it's safe to say that analyzing any historical work requires a bit of back referencing. I believe I watched Spaceballs before watching Star Wars, and I think I enjoyed Star Wars more because of it. Can't say for sure, it was a long time ago.
I'm not from the US, but I have an idea about the amendment's contents. Afaik it makes little sense without the presumption of innocence. So to a person living under the feudal law, the self-incrimination clause would be mostly useless. Similarly, if the mankind adopts hard collectivism in the future (in light of overpopulation), the 5th might become meaningless to them.
Even today, there are countries where constitutional freedoms are just a political circus, or communities living under mob rule, and if you recite the 5th to a person there, they'd laugh at you. In the US, southern towns until ~1960s are often portrayed this way.
The same way, I'm pretty sure the culture ≥2000 years ago was quite different to ours, and personally I'm not interested in cherry-picking the text just to get through it as fast as possible―especially since e.g. Dante or Joyce might have considered different chapters obligatory.