Indeed. The deeper we go, the more we find that separates us from the other Primates. Your main point is dead on: if we and they belong to the same biological tree, the branches are widely separated.
This more recent discovery that during sexual reproduction the germ cell chromosomes bind to their partners with a different method than that used by the other apes definitely raised questions about how we ended up having some of our genes in common.
There are undoubtely ways other than sexual reproduction to acquire genes from other critters. We just don't know them yet. When we do I think our ideas about life's origins will change more than dramatically.