This was a comment, but I realized there is enough to make it an answer.
In a society where the village raises the child, not the nuclear family, there is no need for marriage. It would be a strongly socialist community. Decisions on child raising would not be made by the parents, so there is no need for determining parentage or custody. I am thinking of the Kibbutz community, for instance.
A society with a high mortality rate, for instance, where it is not expected that the natural parents will live long enough to raise the child on their own. Or a very nomadic society. Human babies need a mother for only the first year or so of their lives, but once weaned, nannies are just as effective.
Since marriage is primarily designed to provide monetary support, and establish financial obligations, if the society took care of child-raising costs equally, providing for instance a universal child benefit sufficient to raise the child, there would be no need for such financial considerations.
In such a society, there would be no financial impediment to 'single moms', so I suspect it would actually promote child bearing, not reduce it. There would be no financial penalty to the mother or father, no worry about parental obligations, no worries about consequences (except disease).
Considerations that would have to be addressed:
There would be no inheritance.
Society, not the children, would be responsible for the care of elderly or infirm parents. Japan is addressing this problem, through the introduction of personal care robots.
The contemporary 'Bank of Mom and Dad' would not be available. In Australia, the government (to an extent) replaces parental obligations to set the children up through a lump-sum once-in-a-lifetime social benefit. This, more than anything, equalizes the starting position for all young people, and levels the playing field for opportunity.
Powers of Attorney for Personal Care would be generalized to friends and acquaintances, not family.
Social gatherings would not be family-centric, but social-group centric.
One's fate is no longer determined by rite of birth. All children would be born equal in status, with equal opportunity.
One's sense of 'rootedness' would not be in the family, but in the society and the 'village' that raised the child.
A society that pretty much fits your description and criteria would be the Hawaiian culture before white society and religion impinged on it. See for instance Title: Sexual Behavior in Pre Contact Hawai‘i: A Sexological Ethnography
A makua hine (“aunt”) or kupuna wahine (“grandmother”) did the blowing
(Pukui, Haertig, and Lee, 1972, p. 80). Any number of adult females
was qualified to be the blower for a particular young male, because,
traditionally in Hawai‘i, all age mates of an offspring’s parents were
considered to be “parents” in some way, and all individuals of
grandparental age were considered to be kupuna (grandparent or elder).
Therefore, the same term might refer to a blood relative, a
non-relative, or a neighbor
Individuals stayed together or not by choice rather than by commitment
or obligation. One member of a pair could be monogamous while the
other was polygamous. While public announcements of intentions to stay
together among ali‘i were noteworthy and often elaborate affairs, they
were uncommon. David Malo, an advisor to King Kalakaua III and an
Hawai‘ian convert to Christianity, wrote in 1839: “Of the people about
court there were few who lived in marriage. The number of those who
had no legitimate relations with women was greatly in the majority.
Sodomy and other unnatural vices in which men were the correspondents,
fornication and hired prostitution were practiced about court” (Malo,
1951, p. 65) 9.