not tremendously familiar with Hicks's work, and perhaps some more experienced historians can weigh in on this too, but here are some thoughts about Hicks's method as I see it.

First, Hicks's argument has a few more parts to it than you list in your question. She argues that the sexist worldview of the British civil servants in charge of the computing industry led them to make certain decisions about the workers they wanted and the ways that they would recruit such workers. These decisions turned out to be drastic miscalculations, and the policies that resulted from them starved the computing industry of its needed personnel. The government's response to this labor shortage was yet another drastic miscalculation— the amalgamation of British computing into one company— and that eroded Britain's lead in the computer industry for good.

The crucial mid-level argument here is about employment policy and its effects, and it's definitely possible to establish that sexism underlay employment policies which undercut and then destroyed the industry. Not having read Hicks's book, I can't say exactly how Hicks goes about this, but likely the majority of her evidence here comes from the files of the short-lived British Ministry of Technology, which was in charge of the computer industry. MinTech was established in 1964 and merged into another department in 1970 (which suggests that something big happened in British computing around 1970– departments don't usually merge on a whim). So Hicks likely mined the government files relating to the computer industry to see how British civil servants discussed women workers they were hiring, and how they defined a "suitable" computer worker. The word "suitable" is crucial here. British civil servants loved to talk about suitable workers, suitable nurses, suitable people, etc., and it's usually code for some kind of gendered or classed hierarchy. (Ex: The British civil servants that I study are always talking about "suitable nurses," and in their case, "suitable" is 99.9% of the time code for "middle class.") She read their notes, their internal letters, their inter-departmental working groups, their minutes to each other— how did they define the status of computer workers as a problem, how did they think about women computer workers, and how did they come up with a policy that they thought would solve the problem?

From there she can reconstruct the emergence of a new employment policy and then follow up on its effects— she might have been able to get her hands on employment records, or maybe government ministers talking about how all the young men they hire keep leaving. She can definitely find evidence that the policy, once created, led to a labor shortage, and that the labor shortage led to another, equally disastrous policy.

In analyzing this evidence, Hicks is taking her cues from the work of other feminist labor historians, who have shown comprehensively what happens when women enter traditionally male industries, or how work done by women is usually defined as unskilled and/or "specially suited" for women to do. Example: going back to the 19th century, women workers are often seen as more dextrous than men, and better able to carry out repetitive tasks. After World War I, when women took up positions as clerical workers in huge numbers, clerical work became defined as women's work, with its central tasks being ones "specially suited" for women— repetitive typing, filing, shorthand writing, etc. Another example is nursing. Are nurses skilled practitioners able to decide on and direct treatment, or are nurses simply there to be "the physician's hand" and carry out his directions? This debate has totally fractured the nursing profession since its beginning.

But the above only answers part of your question. There is still this part:

versus so many other things that were going on in the 1960s? For instance, could it have instead been much more important that the US was where transistor technology was developed or that the US government invested heavily into computer research and industry as part of the space race? Can you distinguish levels of causality, such as one circumstance being more important than another or one being essential but another being incidental?

Read a lot of history and you start to see patterns. Most history books will start with an object of study—the welfare state, the Civil War, the computer industry, etc—then they play with it, using one or multiple of these methods (and probably others), based on what’s already been written, what’s missing in present histories, what they think we might be able to learn if we turn the object of study this way or that.

• Juxtaposition: You bring in a second object of study and see what happens when you study them side by side: the rise of the welfare state took place at the same time as the fall of the British empire. Did these two events impact each other? (Yes. The Afterlife of Empire, by Jordanna Bailkin.)
• Periodization: You take the object of study but put a different historical "clock" on it. For example, the reconstruction of France after World War II had its roots not in the liberation of France in 1944 (which is where the clock starts in most histories) but in the policies of Vichy and the Third Republic. (The French New Deal by Phil Nord)
• Renarration I: You take the object of study but study it on a different scale/ from a different place: you study the Civil War as it unfolded in, say, a small town on the Union/Confederate border. Or you go bigger, and you study the Civil War as a global event and examine how it impacted people in Europe, Africa, etc. (Andrew Zimmermann's recent articles on the global Civil War)
• Renarration II: You take the object of study but listen to different actors: this is what Marie Hicks's work does. Her object of study is the British computer industry. But she pays attention to different people— the women computer workers. She re-narrates the story of the British computer industry by telling it from the perspective of women computer workers. Switching this perspective, she argues, changes everything.

Her choice to tell the story of the computer industry through its women workers opens up new modes of thinking— namely, thinking about the computer industry as a site of actual work and labor, and thinking about the laborers who perform it. That choice, as I mentioned above, takes its cues from earlier work on women, labor, and technology. It also takes its cues from today’s Silicon Valley and the role of gender in it. Hicks argues that looking at the historical computing industry from the perspective of women workers—who were pushed out once by employment discrimination and again by their erasure in the histories we tell about technology— can help us understand today’s moment, where once again stories about technological progress and triumph erase technology’s failures and inequalities. Hicks is very clear about what’s at stake in her study and it is that.

Your question hints at another choice, another option: could you tell the story of the British computer industry by going to the US and looking at trends in the US and US investment in computers? Perhaps this would be a story about how the US and the UK were in their own sort of computer race (just spitballing here) and how the UK ended up losing. There’s a third actor here that you hint at, the Soviet Union, so could you ask how the Cold War impacted the UK computing industry? These are all equally valid ways to look at British computing. In some ways your question is not incompatible with Hicks’s— the UK is struggling under a huge labor shortage at the exact moment that the US is doubling down and investing, so the role of the Americans seems to be an operative context here. What would be at stake in making that argument, in telling that story? Why would it matter? What would we learn?

For Hicks it is just context. For you if you were writing a history of British computing, maybe it wouldn’t be— the US might be a much bigger player in your story, and you would pursue different leads and different forms of evidence to bring the US into focus. The resulting two histories wouldn’t be incompatible. They just have different stakes and different valences, and they speak to different concerns. One is a story of British industry collapse from the inside; the other is British industry outpaced by a faster rival.

References for the skill/gendered labor work I mentioned above:

• Anne Phillips & Barbara Taylor, “Sex and Skill: Notes Towards a Feminist Economics” (1980)

• Barbara Melosh, The Physician’s Hand: Work Culture and Conflict in American Nursing (1982)

• Margarete Sandelowski, Devices and Desires: Gender, Technology, and American Nursing (2000)

• Venus Green, Race on the Line: Gender, Labor, and Technology in the Bell System (2001)

An article on historical methodology and the philosophy of science (specifically, the relationship between qualitative research and natural experiments in the social sciences) that may interest you is this:

Kocher, Matthew and Nuno Monteiro. 2016. "Lines of Demarcation: Causation, Design-Based Inference, and Historical Research" Perspectives on Politics 14.4, pp. 952-975.

On the methodology of a particular kind of history (namely, the history of ideas), this is a classic:

Skinner, Quentin. 1969. "Meaning and Understanding in the History of Ideas" History and Theory 8.1, pp.3-53.

A question related to yours, it seems to me, is whether explanations of events that rest on descriptions of intentional actors (i.e., agents that have intentional states -- that is, intentions, beliefs, desires, judgments, etc.) can in theory be reduced to, say, physical descriptions of those events. Here, I think Geertz's The Interpretation of Cultures provides some answers, as do Dennett's The Intentional Stance and scholarship on joint, shared, and/or collective intentionality (such as List's and Pettit's Group Agency, or Searle's The Construction of Social Reality). In light of this scholarship, I think, we can say that there are many good reasons to doubt reductionist explanations of human phenomena. But do historians take sufficient account of chaos and complexity in their explanations (and, if so, how, and to what extent)? I'm not sure. (Edit: My sense is that Tolstoy gives eloquent expression to the idea that the phenomena historians seek to explain are enormously, perhaps incomprehensibly, complex in War and Peace, but I should say here that I know little about Tolstoy in general and this novel in particular. But I think that novel will provide rich food for thought on the question of the epistemic status of historical explanations.)

But, to get back to your question, on identifying causal relationships in history, I reckon that the thinkers who have thought the hardest about this include Max Weber, R.G. Collingwood, E.H. Carr, Marc Bloch, and Fernand Braudel. I suspect, however, that your question remains open.

(Edit: This seems relevant to your question: https://www.cambridge.org/il/academic/subjects/philosophy/philosophy-general-interest/justifying-historical-descriptions?format=PB&isbn=9780521318303)

(Edit: And here's an article that might exemplify an approach to causal identification in historical research: https://www.cambridge.org/core/journals/american-political-science-review/article/examining-a-most-likely-case-for-strong-campaign-effects-hitlers-speeches-and-the-rise-of-the-nazi-party-19271933/1E88D3B79217653D77735294CF2C8DED)