It seems to me that belief in the continued existence of the external world, though enabled indeed by “constancy and coherence” of sense experience, as Hume puts it, is simply an innate, fundamental, and primordial power of human beings. A baby playing peek-a-boo is exercising his power to grasp object permanence.
As for “distinct” existence of objects, it is partially up to us to carve the world at meaningful joints. To do so successfully is yet another power. A measure of intersubjective sanity is required from all people, but still, when a member of a newly discovered tribe points at a rabbit and says “gavagai,” the scientist studying the tribe’s language has good reason to believe that gavagai means “rabbit” rather than “an undetached rabbit part.” The 4 Aristotelian causes play a role here.
So, that there are mind-independent external objects and which objects it is useful to recognize are deliverances of human powers the beginnings of which are built into us from birth.