I don't think its correct to say it isn't studied through that lens at all, but I do think you've correctly put your finger on something here, and I think I should hopefully be able to shed some light on why this is the case. Because this will mostly deal with my experience of Civil War historiography rather than the history itself, it will necessarily be a somewhat subjective post and it might well be that other people will have things to add or disagreements with aspects of it, and I would obviously welcome that!

Firstly I should probably point out that the English Civil Wars (there were actually 3 different ones, arguably more than that when you include the other British kingdoms- see clarification on this below) were by no means a straightforward revolt against monarchy. Many of the king's opponents didn't want to remove him or necessarily restrain the monarchy that much, other than in specific ways relevant to their reasons for rebelling. While there were some on the Parliamentarian side espousing some variant of 'liberte, egalite, fraternite', the majority did not, and were in many cases equally or more authoritarian than the king. Also, in the end, the republic didn't last long, with monarchy being restored in 1660.

That said, all of these things are true to some degree of the French Revolution too, so this doesn't really answer your question.

I think the answer probably lies to a significant extent in the historical dominance of two historiographical traditions about the English Civil Wars, and in the rejection of these traditions in the past 50 years or so. I can only really speak to the historiographical side of things rather than popular culture more broadly, but I think its natural to assume that with events like this that are not hugely well-known in their own right in general popular culture, the broad strokes of academic history will influence public perception.

So anyway, to get to the point, the first tradition I want to talk about is Whig history. This is a term applied in the 20th century to the traditional way in which people in the British establishment in the 18th and 19th centuries had thought about their history . The gist of it is that British history was viewed as an inexorable journey from unenlightened despotism to enlightened freedom. Magna Carta, the Reformation, the Civil Wars and the Revolution of 1688 were all viewed as examples of milestones along this journey.

The Civil Wars had an important place in this way of looking at things, but were actually often seen as less important than the Revolution of 1688. The general view was that while many on the Parliamentarian side were on the right side of history, the Civil Wars were a flawed, violent 'first attempt' at achieving the real goal of constitutional monarchy, which was realised with relatively little violence in 1688.

In the 20th century, historians began to view this whole approach as fundamentally flawed, and sought to distance themselves from 'Whiggism', often downplaying the role of the English Civil Wars as milestones in the development of constitutional democracy in the process.

In the mid 20th century, a new historical tradition came to prominence, led by people like EP Thompson and, particularly with regards to the English Civil Wars, Christopher Hill, which was Marxist history.

This approached stressed the importance of economics and class in history, and framed the Civil Wars as an 'English Revolution' which saw the emerging bourgeoise class overthrowing the old order of absolute monarchy.

The Marxist historians also took a great interest in groups from the period who did display a focus on ideas which could be viewed as proto-versions of ideologies like communism, egalitarianism or liberalism, such as the 'Levellers' and the 'Diggers'.

In the 70s and 80s, there was something of a backlash against the dominance of these views, and a wave of revisionist historians came along who argued for the prominence of religious causes over economic or political. Some argued that it was the king, in fact, who had been the revolutionary, with him and his Archbishop of Canterbury attempting an 'Arminian Revolution' to replace the Calvinist Protestantism that had dominated previously. The Civil Wars, in this view, were mostly about preserving Protestantism, rather than opposing absolutism as such.

I can't claim to be completely up to date with the very latest scholarship on the Civil Wars, because I really last read about this stuff a few years ago now, but basically historians have now become a lot more syncretic about it and will generally ascribe the causes of the Civil Wars to a mix of religious, constitutional, legal, and economic factors.

TL,DR: Two dominant schools of historiography certainly did cast the English Civil Wars as wars against absolute monarchy, but in the case of Whig history it was seen as secondary to the Revolution of 1688, and in the case of Marxist history it was cast as a 'bourgeois revolution' which was as much about economics as politics. In recent decades, a backlash against these schools has seen a much greater focus on the religious nature of the English Civil Wars.