You're asking good questions here. Historians don't try to find sources that are free of biases or show multiple perspectives. The opposite actually. All sources are biased and contain limited perspectives just by virtue of being created by humans, human institutions, and under specific circumstances. How to deal with this problem is the fundamental job of anyone who does research on the past and the basic foundation of the historical method, whether done by a high school student or the most important historian on a particular topic.

Think of the past a bit like a scientific experiment, except the "experiment" only happens once and unfortunately has already been run, and all the surviving data are scattered around the world in a multitude of improvised sites of collection by just whoever happened to have a pen and paper nearby. Your job as the historian is to try and use that data to answer a question that you have about some aspect of that past experiment. In order to do that, you have to gather as much information from the flawed data as possible. Usually, the more data you can find the better, but as you mention, the amount of sources generated at any particular moment depends on a million different factors. But that is all part of the process. Part of being a historian is hunting for the sources of data themselves. How can you actually answer your question? Where might the answer to your question lie? What types of data do you need to convincingly answer your question and say something new about the past?

To make the challenge even more impossible, not only is the data scattered and uneven, it is also fundamentally biased, as you've already noted. So even when you find the data, you can't just use them willy nilly. Now how do you deal with this? Well, it REALLY varies because of the sources. Basically every history book that is one historian's attempt to sort through the biases and limits of their data set to come up with a coherent answer to their particular question.

At the most basic level, you first ask information about each source, each individual data point. Who created the source? When? Why? Did it come from an institution? What is up with that institution? How did it work? Another thing to inquire about is what was happening that led to the source being created. Why did this person create this source? What circumstances led it its creation? What other things were happening at the same time that might explain some of the details in the source? You might also consider how this source confirms or challenges the details of other sources. Is it saying the same thing? Is it adding to it? Is it rebutting it? Does it have a different perspective? If so, why is it different?

In essence, you apply these thought processes to each source, and slowly assemble a data set that can allow you to answer your question. Flaws and all. In theory, the more data you find the better sense of a "line of best fit" so to speak comes into view. This ends up being the narrative of your answer to your question. Of course, it's never as simple as this, but you hopefully get the idea that history is all about using many sources to answer a question. And unfortunately, not all questions can be answered, especially as you go farther back in time. But sometimes, a question can't be answered because it is too big; in order to answer the question you would have to gather so much data of such immense complexity that it can't be done well.

As for how historians will do it in the future, well, that's impossible to say. It's tempting to say they will continue using some form of the basic historical method that I've outlined here, since people have been using it in some form for hundreds of years, but perhaps they will use it in increasingly efficient ways and with the aid of more advanced technology. But who knows! The questions change over time. The things that seem important shift too. New sources come to light. Historians develop new ways of getting data from the existing archives. They find new scientific or archaeological sources which can add more data too. They build off of the findings of others to ask new, previously unimaginable/unanswerable questions about forces that weren't visible to people who lived through historical moments. They get new technology too. 100 years ago a discussion forum on the internet about history was unimaginable, so what will it be like in 100 years from now? Maybe quantum comping will somehow change the whole game. They'll probably develop methods we can't even imagine. Either way, I am looking forward to smart people telling me more about this wild period I'm living through.

To give the usual suspects' inboxes a break, I'll link to the last time I linked my usual response to this question.