I have a lot of respect for articles written on thetorah.com . Having said that, a lot of articles on thetorah.com do not fit within "Orthodoxy" as we currently have it. Orthodoxy considers that the Torah was more or less given as a complete unit either by the time Moses died, or by the time that Joshua died. The article you cite does not believe that the Torah as a book was given as a complete unit, but rather the Torah is a series of edits or layers that were eventually merged into a book that we now have as a complete unit.
The reality is we have zero concrete proof for either argument. Even with a close read of the Biblical account one can come away convinced of both arguments, that either the Torah was given as a complete unit, or that it was given and edited in stages. For example the books of Ezra and Nehemiah are convinced that the Torah they have is a complete unit given by Moses. But when you read the book of Kings and a copy of "the Torah" was found after some construction at the Temple, and somehow the current priests and kings can't tell if what's in that Torah is legit or not. The only way that can be the case is if the Torah they currently had was incomplete, or they didn't have a surviving Torah at all.
So if someone chooses to believe that the Torah was given as a complete unit, they won't discuss the priestly or Deutero authors, they will discuss the "Priestly voice" such as Rabbi Lord Jonathan Sacks does in his book on Leviticus. And they will view all priestly material as speaking from the viewpoint of the priests and during these Torah sections the priestly voice is just one of many voices reflected in the Torah.
If someone chooses to believe that the Torah was compiled over stages, then there become serious implications. The Israelites in the desert didn't circumcise their kids because they didn't know they were supposed to.
Personally I don't think either of these situations is "the truth" and that reality lies somewhere in the middle. For example, I can completely believe that Moses received the Torah as a complete unit, but then Moses elected elders and judges and had to hand over that Torah for them to adjudicate and decide law. The Torah was always a framework to be given over to Moses and the judges to use and adapt to ongoing situations. Therefore it wouldn't surprise me that after a few generations these different uses of the Torah needed to be tied back into the original source and put together as a unit at some point. And so I don't view the Torah as a series of edits, but rather a series of legitimate collections of the Torah in the different communities as they received it. The Torah according to the Levites/Priests after a few generations is going to be it's own distinctive voice as they will focus on the laws of holiness, sacrifices, purity, etc. To say this "source" is just a series of edits doesn't do justice to the legitimacy of the source.
For the question of whether the Torah was a complete unit or not, you will have to decide for yourself. But even if you do decide to accept that the Torah was not given as a complete unit, I would encourage you to not immediately jump to the conclusion that many atheists do that the Torah is then somehow "fake." I don't believe that anyone at any point in Israelite history viewed the Torah as fake, even when they had multiple different copies of it in their hands. The Essene Community in Qumran had collected many different versions of the Torah and they did not believe that the different versions meant the Torah was fake. Rather they saw all the different received traditions as a window into the Torah and its truth.
If you are looking for a book that addresses the single question of the Torah as a complete unit vs a compiled book, you can read Torah from Heaven by Normon Solomon.
If you decide to accept the idea that the Torah and possibly other Biblical books could be compilations, then I highly recommend the book How to Read the Bible by Marc Zvi Brettler. It presents the known textual evidence and archaeological record in a very accessible manner. So far it's been one of my favorite books on the topic.
It's worth noting that while I read and support all honest attempts at understanding the Bible and its history; I find the Documentary Hypothesis to be lacking at times and worthy of being challenged. For example they divide the texts into "sources" but now they are arguing over the sources and whether there should be more or less sources. And often times they make grand assumptions like saying this part of the Torah doesn't know what's happening in some other part, but the reality is the Torah comments on so much in very subtle ways that these secular scholars don't notice because it's not obvious enough for them. I heard someone say that the Israelite Children who were thrown into the sea were not known by the rest of the Torah since this ethnic cleansing isn't "brought up again in the Torah narrative." But such an idea could not be farther from the truth in my opinion.