Forgive me, as I am but a graduate student in an interdisciplinary social science and not a historian. Just as countries' economic histories are diverse, so are the theories (and contradictions between and within them).

However, a very general sort of story that is widely accepted among economists and political scientists (with of course some variations and disagreements) is that of Douglass North (see his book Structure and Change in Economic History, and Institutions, institutional change and economic performance). The basic idea is that the social, legal, and political "institutions" are the primary determinants of economic growth and change. "Institutions" in this literature has the simultaneously specific and vague meaning of "rules" that constrain and incentivize certain individual and social behaviors. Such rules can be formal (as in written down and enforced with the mechanisms of the state police power) or function more informally as norms, with punishments and incentives operating more at the level of social relations.

Taking those vague ideas as given, the question development economists are wrestling with is, what sorts of institutions lead to "good" economic development pathways? Some types of institutions that political economists seem to agree have something to do with economic growth are those protecting property rights, and those surrounding the form and enforcement of contracts between economic actors.

Property rights are important, the story goes, because people will not invest in maintaining and increasing the value of their property if it, and the product of their labor, is at some sufficient risk of being expropriated by their government or other people.

Contracts are important, because they allow people to essentially make deals with each other for mutual gain. The idea is, without strong institutions that guarantee contracts, transactions and trades between people are discouraged because everyone has strong incentives to cheat the other person.

Doing actual research on this is hard, especially in terms of figuring out your "necessary" and "sufficient" question, as well as the related question of causal relationships. The development economist Daron Acemoglu and those who work with him have done a lot of quantitative work trying to suss out what kinds of institutions cause economic development. See Acemoglu, Daron, and Simon Johnson. “Unbundling Institutions.” Journal of Political Economy 113 (2005) for an example of the type of study I'm talking about.

Of course, a big difficulty with your question is that if you actually try to go back and do deep case studies of countries as they industrialized, it's not really clear if property rights and contracting institutions and other institutions caused economic growth, or if industrialization, through associated political processes, caused those institutions to be instituted.

I've read a couple of development economics books where the author basically says, "We don't really know why some countries do well and others don't". Is that really true? There must be at least some core criteria - "necessary but not sufficient" kind of stuff.