I don't think anybody's recommended specialist texts so far. As I've been at some pains to try to make clear in each post so far, the notion of a yawning gulf between "two-page summary in the middle of The Economist" and "scientific journals" is as utterly false in history as it is in any field. Mouthwash has been exaggerating fairly dramatically.
Let's go back to the Halsall example. The book (Barbarian Migrations and the Roman West) is a relatively recent textbook, about ten years old. It is primarily based around a narrative of the last century of the Western Roman Empire, combined with segments on thematic history and overviews of the Way Stuff Worked Back Then to further explicate things. While it contains a brief and non-exhaustive overview of some recent historiography, and stakes a claim in a fairly complicated debate, it is undoubtedly not a specialist text. The reader does not need to know very much about late antiquity to be able to use it effectively. That's one of the reasons Halsall spends so much time describing stuff like how identity works, and it's why he even includes a narrative at all. Specialist academic history rarely contains narratives, and certainly does not contain a narrative of more than a century of history. Certainly nobody has pointed to complicated journal articles, or texts that engage even more directly with specific things. Fundamentally, if somebody wants narrative history, they are looking for a relatively general textbook, and that is what they are going to be offered.
Even authors who write "relatively general textbook"s in history, however, find themselves having to engage with other authors. That is, after all, what the field is. It includes a great deal of collaboration and also a great deal of disagreement over interpretation. I would say that, while it's important to avoid taking it too far, interpretation assumes a somewhat larger role in historical academia than it does in STEM fields. There is no one right answer about why something happened, although there are plenty of wrong answers; while the field is fundamentally based on facts, it is fundamentally not based on experimentation. It is about intersubjectivity and persuasion. You can't avoid having to talk about multiple historians' opinions on a given subject, unless you are trying to give the impression that a field is settled when it is not. I will agree that Barbarian Migrations spends rather more time on historiography than some other such narratives. I believe, however, that it is a relatively small price to pay for what I believe is the best and most accurate narrative of the relevant events, as such things go, and I also believe that the interested reader will be able to make it through without much difficulty. I do not believe it is a perfect text; there is certainly room to update it, and editorial decisions I might not have made. There are some competing recent texts, which Halsall actually references in the section that Mouthwash complains about. He and anyone else are welcome to try them, although I believe that they are all flawed in more serious ways.
The journal article comparison doesn't work for another reason. With relatively few exceptions, the sciences that most people know and are taught in school are more or less settled. Nobody is going to seriously question the theory of evolution, or say that hydrochloric acid doesn't actually exist, or declare that free-body diagrams aren't worth using. But the history that is taught in school is constantly reevaluated to the extent that what is taught in middle and high school is often wrong - not just "incomplete" or "not the whole story", but flat-out wrong. In history, there is rarely such a thing as a settled field. There are not only explanations that have been discarded by the academy (for example, Gibbon's explanation of the last millennium of Roman history has about as much persuasive power for modern academic historians as the theory of phlogiston has for modern chemists) but a multiplicity of competing, possibly-valid explanations that are current and have not been discarded. A historian cannot begin a discussion of the fall of Rome without having to account for all the people who believe that it happened because of Christianity, or because of Iran, or because of "the barbarians", or because of stirrups, or because of lead poisoning, or because of inflation. It makes a great deal of sense, then, for a good historian to spend at least a little time in her general narrative history discussing these things and why they are not actually Things. This discussion might take the form of referencing the work of specific other historians, and it might not, but its fundamental purpose is still the same. And it is still quite far away from the sort of hyperspecific, highly detailed, and deeply complicated exegesis that one might find in an academic journal. One might compare it to having brief discussions of the work of Ptolemy, Copernicus, Brahe, Kepler, and Galileo in an introductory text on astronomy.
This problem, that a field is not settled and that even basic questions remain matters of serious contention in some ways, is one of the sources of the problem. I have repeatedly alluded to the hospitalitas debate as one of the more complicated segments of the Halsall text. The debate is fundamental to the field, because it addresses the actual mechanism of settlement. Pretty much everybody agrees that, in the fifth century, a bunch of people moved around the Western Roman Empire, and some people who didn't originally live in the Empire moved into the Empire, and then the Empire stopped being an Empire (without assigning any particular amount of causation to any of those things). So figuring out how these people settled is a pretty important discussion to have. It gets back to Mouthwash's original thing: how did any of this stuff actually happen for real people, anyway? Did barbarian hordes swarm into the country, plundering and pillaging, destroying local government and putting up something new in its place? Were these people not even really barbarians, but the imperial government's own soldiers, quartered on local land in a time-honored Roman practice even after the Roman Empire stopped being a Thing? Was the whole thing a matter of transacting taxes, not land? Was it none of these things, or all of them to some degree? Historians clearly have a lot of opinions on the matter. However, due to the nature of the evidence, most of the discussion centers around legal codes devised by post-Roman rulers in a highly awkward form of Latin. (There are other forms of evidence, but the legal texts are decisive.) The debate is fraught, contentious, and almost opaque to all but about maybe two hundred people in the entire world. You can't entirely ignore it, especially not in a text that purports to be a summary of history, when it was one of the most important aspects of political and social changes in the fifth century. I think that, in this particular case, the author made the best of a bad situation and provided an adequate summary that takes a stance on the issue without going too far into the weeds. There are other books (e.g. Walter Goffart's Barbarian Tides) that discuss the debate in rather more detail and thus have a greater likelihood of getting the non-specialist reader lost.
But that might be providing too much detail. Mouthwash has already said that he thought the first dozen or so pages of the text were what turned him off of it. This is a bit surprising to me, because many of those pages are taken up by three brief example case studies of Interesting Stuff That Happened In The Fourth Century that most history books use to help introduce fourth-century Rome to new readers (except for the last, which is the same thing except for the sixth century); some of the following pages are a thumbnail sketch of historiography, but nothing in the sort of detail that I'd be worried about offering to anybody who is actually interested in the material capable of reading beyond a high school level. He can't not talk about it, because a big part of his thesis is that common ways of seeing "the barbarians", e.g. as a rampaging horde, or as the kernel of modern civilizations, are flawed. Again, this gets at how things actually worked for real people, but gives a brief sketch of opposing viewpoints.
Mouthwash is, of course, within his rights to regard the book as overly complicated or boring. I probably wouldn't recommend it to most readers not yet in undergraduate studies, and if he is one of those readers I apologize. But it undoubtedly has a great deal to do with his other interest - finding out about real stuff - and although it might be a bit easier to absorb, it could undoubtedly be much, much harder.
