while the potential social utility of religion might not depend on it's "truthfulness", the actual ability of religion to provide benefits to society does depend on it's "perceived truthfulness" to some degree.

taking myself as an example, I'm not religious, never have been. in fact, I'm fairly convinced that it's all just mythology. no matter how good of a case you could make for the social utility of a religion, it's unlikely to convince me that the religion is actually true. that limits the social utility of the religion because at least one member of the society has a high barrier to adopt it as a personal moral guide or philosophy.

in this regard, the social utility of religion does depend on it's truthfulness (or at least, on its ability to appear truthful). an untruthful religion is necessarily fragile in its capacity to permeate a society homogenously and persistently, because once the seeds of doubt are sown, the validity of the religion as truth will always be questioned, and it therefore is unlikely to be fully adopted by everybody in the society. a society with various different religions, and a mix of religious and non-religious groups will not gain the benefits of a society that has a single religion.