Could society function without religion?
I suggest you will never find a society with zero religion without suppression of religion (which as communism has shown only makes it go underground).
However, if you accept that a society that has the vast majority of its population not believing in a personal god as “without religion” then Denmark and Sweden meet your requirements as a test case.
Although the vast majority are still members of the Lutheran Church (it has been the default position for decades, you have to actively remove yourself), many still get baptized, and many (if they get married) still do it in church. Through poll after poll after poll and indepth interviews these societies are as openly and voluntarily irreligious as any society that has every existed. Most as stated, do not believe in god, but if the do still have anything to do with the church are so called secular cultural Christians, just as many jews are cultural jews. The beliefs in Jesus as the son of god, and heaven and hell, and salvation and original sin are gone for the vast majority, but the ceremonies remain and are seen as nice traditional aspects of their culture.
These societies come in the top 10 for all common indictors of societal wellbeing, be they peace, lack of corruption, low murder rates, high foreign aid, universal health care, happiness indicies, etc..etc.. etc…
Could a society function without religion? They can and they do.
Furthermore, the prevalence of people who are not just irreligious, but ambivalent about the god question and life after death, challenges the common perception that religion is some how a biological necessity for people.
I suggest those of you think otherwise might gain from reading “Society without God” by Phil Zuckernman.