There isn't really one "scientific method." There isn't one today, and there was never one historically. It is a myth that we teach children in grade school, so that they can structure their thinking a little better, and gain an appreciation of the value of science. In practice scientists use a wide variety of methodologies, and always have.

But if you are asking specifically about the kinds of transformations in knowledge-production (which have some aspects of "method" but are largely also about customs, cultures, and social arrangements) that took place in 16th-18th century Western Europe (aka, the "Scientific Revolution," which most historians of science don't like as a concept either, but we'll ignore that for the moment), then we can talk about the conditions that led to that.

Was it inevitable in an agricultural society? I see absolutely no reason to suspect it would be. For one thing, there were thousands of years of agricultural societies that did not produce these changes, and indeed there were very complex, bureaucratic, agricultural societies other places in the world that did not produce all of these changes (Asian societies, for example — there was no Chinese or Japanese "Scientific Revolution").

Was it then an "accidental product"? I don't really know what you mean by "accidental" — I wouldn't call what happened accidental, myself. But I would say it was the result of very long-term cultural shifts that were contingent, which is perhaps the same thing you mean. The changes of the "Scientific Revolution" period did not come out of nothing and were not as an abrupt shift from what came before as people tend to believe (hence historians of science not liking the term), but there were some changes and they were the culmination of many thousands of years of history. Some of these were very specific to knowledge-production, like influences from Arabic and Greek scientific writings, while some of these were very broadly cultural, economic, religious, and so on. There's no single reason for these things working out the way they did, and it wasn't an overnight thing.

A better way to think about this than the framework you have is that what we call Western Science was something that emerged as a specific approach to knowledge-production and circulation over the course of many centuries, and was a specific type of culture by itself. It was a culture that emerged out of other cultural influences, like all cultures. Those cultures owed much to the specifics of Western Europe in the medieval and early modern periods — its political landscape (fractured and competitive), its religious ideologies (which often emphasized knowledge of nature as a means towards knowledge of God), its religious institutions (which often sponsored scientific work and harbored knowledge-producers, like the Catholic Church), its military/political ambitions (war, colonialism, imperialism), its economic situation (wealth concentrated in patrons), its philosophical traditions (a spirit of individual inquiry going back to the Greeks, its glorification of Platonism and Aristotelianism, both of which in different ways influenced its approach to the study of mathematics and nature), and so on. "Accidental" is the wrong word for it, but it was no more inevitable than, say, Western European musical culture, or Western European philosophy.

To take this approach de-centralized "science" as something separate from culture, which many people today (notably scientists) find disturbing and uncomfortable, in part because of the aforementioned stories we tell children to make them more accepting of science (we like to set it up as something separate from the mucky human world of politics, economics, culture, and so on, in order to emphasize the transcendent, objective nature of its conclusions). But this is how historians of science (and anthropologists and sociologists of science) see it: a human activity, a form of culture and society, one bent around the practices of knowledge-production. Western Europe exported this particular cultural form globally (among other exports) through various means (some violent, some not), and today it is something of a global culture (it is not longer exclusively the provenance of Western Europeans), but that does not make it any less cultural.

A great primer to how historians of science see the "Scientific Revolution" and science in general is Steven Shapin's The Scientific Revolution (University of Chicago Press, 1996).