I suspect the reason you haven't had many answers to this question is that it is an enormous question that deals with the philosophy of history, historical theory and historical methodology. All of which are even individually massive topics that are incredibly difficult to summarize in a reddit post. So, I'll do my best to introduce you to these ideas, but it will be impossible to explore them fully and it is also possible that you won't find the answers completely satisfying. If you have follow up questions, I'll try my best to answer them.

I will start with a restatement of what I see in your question, so that you understand the perspective from which I am answering. You are chiefly concerned with, as near as I can tell, what might be called incontrovertible fact. That is to say, you are interested in a narrative of things and people and events (all broadly conceived) that might be able to be called something like objective history. I am going to make a fairly simple assertion here (and hopefully prove it to you over the remainder of this post). There is not a single, unerring, "correct" version of history. At the same time, there are not an infinite number of "correct" or reasonable versions of history. Wrestling with this fact is much of what historians are doing. To wit:

Your concern about the independence of sources is certainly not unwarranted. Historians deal with precisely this issue quite consciously when engaging with sources. Who wrote it? What was their motivation or perspective? Why did they write it? Replace "write" with the appropriate verb for non-written sources, of course. These are all common and indeed vital questions that historians use to interrogate their sources. They are also derived from the understanding that sources are not, in a pure sense, independent. They are, of course, dependent. Many of the most important insights you can gain from sources are not necessarily explicitly contained within the sources themselves. The notion of "reading against the grain" is absolutely vital to getting the most you can out of historical sources, as a historian. Reading against the grain means, to an approximation, not just reading for comprehension of the text, but interrogating its origins, how it constructs meaning, what assumptions are being made by the writer, and so forth. That doesn't mean that you can never rely on what is explicitly present in a source, but rather than you can't assume to rely on it. But we can still draw real conclusions, to choose some examples of wildly different importance: The holocaust really happened. The Soviet Union really did disintegrate. The US really did drop bombs on Hiroshima and Nagasaki. Elvis Presley really did die. And they really did make another Transformers movie a few years back (please stop!).

Historical sources, therefore, do not speak for themselves. Historians speak for their sources through interrogation and analysis. Historians build a narrative by consulting many different sources from as many different places as they can find. When things appear in more than one place in the historical record, the evidence for them is greater, for example. However, that is not the only standard. Our methodologies (like reading against the grain in archival sources, or questioning someone effectively in an oral history), as well as our frameworks for understanding things, aka theories that inform our analysis like gender, class, and many many others, help us to evaluate those sources and their usefulness, as well analyze them.

Historians, first and foremost, I would say - are making arguments about the way things were, they are not chronicling the way things were. It is, then, ultimately not the single historian or single work of history that provides "the truth" about history that you are looking for, but rather the field itself in its struggle to understand the past. So while historians are obligated to try to be unbiased and dispassionate in their analyses to a degree, they are not simply bound to saying nothing about the history they are studying. Quite the opposite. You can be both bound by your evidence and quite impassioned about the subject. Indeed, much of the best history has precisely this combination of motivations behind it.

History isn't, then, made up of a bunch of a discrete facts, people, events and so forth. It is much messier than that. Here, I'll borrow John Lewis Gaddis' words: "... if you think of the past as a landscape, then history is the way we represent it, and it's that act of representation that lifts us above the familiar to let us experience vicariously what we can't experience directly: a wider view." (Gaddis, Landscapes of History, OUP, 2002, p. 5). Of course, several other things follow from this understanding of history. The first one that comes to mind is precisely what I mentioned in the first sentence of this paragraph. Put otherwise: history is not reductionist. I think there is a bigger philosophical discussion to be had here about reductionism, but we can only talk about so much in one post. Suffice to say: reductionism, or breaking down history into component parts of people, events, actions, all of which have a precise and identifiable series of causes, is not a particularly fruitful way of understanding the past. Such an understanding of history would be, I think, precisely the opposite of Gaddis' understanding of "the landscape of history." If you are interested in this idea (and I think it speaks to your concern about independence of historical sources) Gaddis tackles this question in detail in Chapter 4 of the above-cited work, titled "The Interdependency of Variables."

So, history is messy. We do our best, we struggle with our sources, with our analyses and with our theories. Learning to be comfortable with ambiguity while still making forceful arguments about the past is one of the most difficult things about learning to be a historian, in my opinion. Again, I return to my assertion that just because there might be more than one interpretation of narrative that could be called correct, it does not mean that there are infinitely many. Our ideas are ultimately put into that being roiling pool of historical discourse, and if they stand the test of open, rational discussion, then we call them good ideas. But there is no "end" of history. There is no time when we will have gotten it all right, when we will have mapped the entire history of our species from start to present. Not only will it not happen, it's not even, I would argue, the point. History is, as another historian E.H. Carr put it, a conversation between the present and the past. Insofar as our present it always in flux, our conversation with the past will continue to develop. It is a conversation that can be messy and contradictory. If I can be permitted to quote Gaddis one last time:

And what does all of this have to do with the landscape of history? It's simply this: the possibility that historians may stand, in their relationship to the past, in something like the position states do in their relationship to territory and society. For in "mapping" the past, the historian too is laying down a grid, stifling particularity, privileging legibility, all with a view to making the past accessible for the present and the future. As is also the case with states, the effect is both constraining and liberating: we oppress the past even as we free it." (Gaddis, 135).

Historians don't take sources at face value. A lot of the work of a historian is to account for bias (both deliberate and circumstantial) in sources, place these sources in the right context, and to compare them with other sources. I for one am deeply skeptical that there even is such a thing as a truly independent source, no matter when it was written. No honest historian would base a work on a single source and claim that it was in any way, shape, or form "correct". At most, you'll find them talking about how the conventional view of history is fundamentally challenged by this new source and must be changed.

That leaves aside the very notion of historic "truth", which is problematic in itself. We only get a small part of the available evidence - the rest was destroyed in the time between the event and today - and so we can, at best, get an incomplete picture of what happened. When people do talk about undeniable facts in history, they are talking about something that has enough evidence backing it up, and not enough evidence contradicting it, that denying it runs into severe problems. In that way, it is not unlike all other natural science - new theories based on new observations might come up later, which will make the current theories obsolescent (such as Newtonian physics being superseded by relativity), but that does not make the current view any less "true".