Girard claims that the single victim mechanism ending in divinization of the scapegoat is the foundation of all cultures. He sees, the world over, in many genesis myths and divine figures' origin stories, deeply encoded collective violence – the figure is murdered (Abel), marries in a sacrificial way (Persephone), is dismembered to make something (Ymir), is consumed (Dionysus), etc.* Societies organise themselves around this ineffable experience of transcendence – this is what brought us together the first time, so this is what defines us.
I, personally, amateur that I am, think that sometimes Girard overstretches himself to make a very good theory fit absolutely everything. I can see this being the origin of organised religion, perhaps, or capital-S Society (the intangible structures by which people organise themselves), but culture? Maybe, being an artist, I am thinking about culture too materially, but what does mythology have to do with the shape of your spearheads, or your recipe for bolognese? Chimpanzees have cultures, without having mythology.
As far as religion, I would posit that – independent of mysticism – its ultimate beginning grains are in superstition, and
even pigeons can be superstitious. There is enough natural inclination towards superstition in humanity that weird pseudo-religions pop up all the time, for example
Monty Jesus or cargo cults or 'cursed' objects that, if they are disturbed, will somehow bring you bad luck ('we don't know how it works but just don't touch it, OK?') which I'm sure everyone has encountered at least once. There is something in us that wants to propitiate the random fluctuations of the space-time continuum to randomly fluctuate in our favour, or at least not against us.
However, I can definitely see the single victim mechanism playing into, and reifying, a nascent superstition. You get a bit of a culture going and into the mix of The Way We Do Things is whatever superstitious practices have arisen organically. Then the magical unifying, purifying effect of the single victim mechanism happens and you think,
wow, whatever that was, it really worked. The divinised scapegoat is very easy to graft into your existing superstition, and then you're on your way to a mythology and, out of that, religion.
*Girard quotes stories that don't come from Europe and the Near East too, I just can't cite them off the top of my head or find sources online for you to read, so I've gone with the more commonly known ones. The radio series uses The Wife of Python, from sub-Saharan Africa, to decode collective violence in mythology. I remember reading a Salish story about a woman who was married to a whirlpool to stop it drowning fishermen, which had collective violence written all over it, but Googling has turned up nothing, alas!