Sheikh M.Kashaf McIqbal certainly answers part of the question very well.
Certainly cooperation is one if the fundamental aspects of our success (and is fundamental to understanding the evolution of complexity more generally), and Axelrod showed long ago that raw cooperation is vulnerable to exploitation, and so requires attendant strategies (that detect and remove cheating strategies) to succeed (at each and every level).
So yes, cooperation is one critical aspect of our evolutionary success (and prerequisites for our future survival), and it is more complex than that, and involves aspects of tool development and use, and aspects of the development and transmission of new levels of abstract communication and patterns and algorithms and the development of new levels of networks.
And the question can go deeper into different dimensions, like the degree to which any of us get influenced by things we are not consciously aware of, as against the degree of influence that consciousness has. That question is deeply debated in Artificial Intelligence, philosophy, psychology, theology, economics, politics and various other disciplines.
How much do we influence, and how much are we the tools of patterns that we barely notice let alone comprehend? And certainly, if one combines those two, then we and our various levels of systems and technologies have certainly significantly altered the relationships of most macro biological systems on the planet (things big enough to see with the unaided eye); our degree of influence on the microbiome might be much less in many contexts. Our influence over earthquakes, volcanoes, plate tectonics, and deeper geophysical processes would be very close to zero.
I would say we are now a significant influence on macrobiology, and an influence on meteorology, and “rule” seems to be too strong a term (a hubristic illusion containing existential level risk).