That's an interesting question. I guess I should start by saying that the way I look at religions preferences the present over the past--I'm a lot more likely to consider how the Church of Satan manipulates, bends, and re-articulates the image/idea of Crowley to it's own ends in the 1960s when it formed than I am to consider how the teachings of Crowley in the 1930s would eventually transform and combine with other streams of thought to produce LaVeyan Satanism if that makes any sense. It's less about drawing a continuous line from the past to the present and more about looking at how the past is re-invented as a resource in the present.
I suppose the three "key elements" I would cite are skeptical rationalism, hedonistic objectivism, and individualistic romanticism. Looking just at atheist/ autotheist forms of Satanism (though I think this could be expanded to somewhat contextualize things like the Temple of Set), the first and most important break from dominant, Western religion (ie: Abrahamic religions based on faith in God and obedience to His laws) is the rejection of a transcendent, supernatural entity/ realm from which sacredness, meaning, and thus guidelines for ethical conduct flow. LaVey instead located the sacred in man's carnal urges and needs, leading him to an understanding similar to Ayn Rand's objectivism that heavily stresses hedonism.
Both of these (skeptical rationalism/ hedonistic objectivism) tie into what I would probably argue is the most important intellectual predecessor to modern Satanism--existentialism. More than anything else things like the Church of Satan seem to me to be an existential backlash against mainstream religion and society's notion of stable meaning and purpose rooted in divine truth. In a lot of ways it then seems sensible for the aesthetics of Satanism to be so deeply rooted in individualistic romanticism, which has a long history of the individual turning his back upon society to seek the exotic, magical, dangerous, and primal; more than one scholar has noted that the modern-day CoS would probably be impossible without Milton's Lucifer.
You could easily fit individuals and organizations like Crowley/ the OTO and A.:A.: into some (or several--they're pretty broad and abstract categories) of these overall movements, but those are the big ones that I would look at. Trends in (sub)culture(s) moved away from rooting one's sense of meaning/ purpose in a transcendent God and the overarching supremacy of civil society based upon restraining natural urges because of religious ethics. Manifestations of this like anti-Enlightenment romanticism and (particularly Nietzsche's) existentialism supported the idea of making the passionate individual the new source of meaning/ legitimate authority. The result was that at least some individuals, from Crowley to LaVey to Long/Myatt, found that the society they were in imparted them with values radically opposed to those of the dominant religions, and chose to respond at least in part with social critiques expressed in starkly transgressive, oppositional religious doctrine.
