Religion is a variety of philosophy that generally is associated with the belief and reverence for theological concepts.
I make a stronger distinction between philosophy and religion.
Religion is more closely associated to
myth than to philosophy in my opinion, where 'myth' doesn't mean 'bullshit', but rather something like 'explanatory account presented as a narrative in the form of a story'. Philosophy is distinguished from that by its attempt to use rational argument and reasoning to justify everything.
For example, religion historically accounted for the origin of everything with creation stories (like the very abbreviated verses at the beginning of Genesis) and by
theogonies. The latter were more common and prevailed where different gods and goddesses were associated with (and served as personifications of) different aspects of reality. So the elaborate stories about which gods and goddesses gave birth to others, and what kind of conflicts and strife arose among them, represented early (more or less unconsciously allegorical) attempts to account for how the various aspects of reality relate to one another.
The earliest Greek philosophers in Ionia addressed the same issues, the origin of everything and how various aspects or reality relate and interact, but didn't personify them and tried to relate them using reasonable arguments.
That use of reason to address foundational issues is more of less definitive of philosophy in my view. Science didn't exist yet in ancient Greek times, though trial-and-error craft traditions certainly did. And until the 1800's science was most commonly referred to as 'natural philosophy' (that specialty of philosophy that addressed the natural world, alongside other specialties like logic, epistemology, ethics and aesthetics). So our distinction between philosophy and science is comparatively recent.
Anaximander imagined the original 'stuff' ('arche') of the universe as formlessness (his 'apeiron', 'without boundaries', the 'undefined'). Then he tried to imagine how it might have acquired definition and form (setting in motion
the form/matter metaphysics which is still very much alive today, in the form of things like
structural realism).
The Pythagoreans (Pythagoras seems to have been an Ionian who moved to southern Italy where his school was influential) used simple mathematics such as numbers as its explanatory principle, where mathematics more or less plays the role of Anaximander's perimeters, shapes and forms, and we still see something of that in theoretical physics today.
Atheistic spiritual authority? A shaman seeking knowledge through divination of animal parts is a prime example of theistic behavior, and hardly comparable to a mathematical analysis of strategic behavior by a modern ecologist. Now if the ecologist was also a practicing numerologist you might have a point.
Like theoretical physicists with the equations that they scrawl all over their chalkboards and believe explain the deepest mysteries of the universe?
I think that an ambiguity in the meaning of 'atheist' is exposed there. If 'atheism' means 'disbelief in the existence of gods' (including the big one the monotheists envision), then it's entirely possible to be spiritual without being a theist. (Imagine Theravadan Buddhist monks for whom gods may or may not exist but are irrelevant to their practice. The gods, should they exist, are probably more in need of enlightenment than we are.) But if 'atheism' is interpreted to mean 'non-religious' or even 'anti-religious' (as it often seems to be by the 'New Atheists') , then you might have a better point.
And I think that 'shamanism' might be best conceived as the belief that there is a higher spiritual world besides this one, and that the higher world can be contacted and influenced by ecstatic means (by altered states of consciousness). We supposedly all experience something of it in dreams. I think that it's entirely possible (and probably most common historically) to conceive of such powers and abilities among specially endowed people without belief in the existence of named and personified gods. There's just something powerful on a higher (abstract?) plane (the laws of physics?) which influences events down here in the world we inhabit. (Of course physicists don't typically believe that they can influence the laws of physics by psychedelic means, though some of the quantum idealists (and their intellectual ancestors, the Kantians) seem to come very close with their belief that consciousness constructs physical reality.)