Is it true that we didn't even need to invent religion(s) until after we had become an agrarian society?

I think that really, really depends on your definition of religion.

Ancient burial sites have been discovered with worldly possessions buried with the dead. I want to say that it was a Neanderthal site where bodies were thrown into a pit, along with hand crafted weapons (I think I’m mistaken about it being Neanderthal, however.)
That is not solid evidence that early humans necessarily believed in a deity, but it could raise a few eyebrows at the very least.

Googling to back this up, by the way…

EDIT: Alright, I found it. It was Homo heidelbergensis that had burial pits and evidence of ritual behavior, such as burying handaxes with the deceased and what appeared to be ritualistic killing of animals, such as horses.

Foragers (hunter-gatherers) today have ideologies similar to religion. the native Americans were largely foragers and they had religions.

So, no.

Agriculture did allow people to form complex societies—and complex religious hierarchies.

No; we didn’t need them as hunter/gatherers but I believe we wanted to believe in gods or a god.