Yes, the great apes branched out from a common ancestor and, given enough time, may branch still further into new species. "Enough time" being the operative concept.
As for links, fossil evidence is notoriously rare and difficult to find, but it is accumulating. There will always be gaps, of course, and each new find creates two more gaps.
There is also genetic evidence, though, which is much more robust.
You continually bring up objections to established science, but what alternatives would you propose -- magic poofing?