If the Buddha existed and books were written about his life and teachings by people who would have a definite knowledge of these, how can we demonstrate any historical link between these writings and the life of the Buddha?
We cannot know for sure. In any case, historical links are irrelevant in terms of early Buddhism.
How come there are so many contradictions between the different Buddhist traditions? To the point in fact that no original idea of what he actually taught seems to exist.
There are suggestions in the early Buddhist canon that false Dhamma would arise ~500 years after the passing of the Buddha. This seems to be the case to me, as Mahayana supposedly arose around that time period.
How come the texts we have about the Buddha date centuries after the supposed life of the man?
The disciples who reached the consummate goal - the Arahants - supposedly perfected themselves to the point where they achieved the iddhi of the "divine eye/ear", which allows them to observe the past for themselves as if they were there, just as we can observe things around us in the present with our mundane eyes & ears. With the divine eye/ear, they are said to be able to recall the Buddha's past words as if they were there witnessing them in person. Supposedly, some of these Arahants recorded the Dhamma in written form as the Buddha-sasana (500 years) was coming to an end, in order to preserve as much of it as possible so it wouldn't be completely lost.
How can they be considered historically credible given this distance?
Its historical credibility cannot be certain, nor directly & personally verified, until we ourselves achieve the divine eye & ear to witness these things for ourselves. However, when I read the early canon, I hear one voice through all of the thousands of suttas, so I have no problem believing that they truly originated from one mind.
"Historical credibility" is the wrong question to apply to Buddhism.
The Buddhist Way - as I understand it - is not based on any alleged traceable
historical credibility, but on
personal applicability in the here-and-now. Historical credibility is only a question applicable to religions which requires that their disciples maintain a faith in the actuality of non-personally-observable historical events. This is not the case for early Buddhism.
All of the ancient Buddhist
stories - e.g. about the Buddha's life - could very well be false and possess zero historical credibility, but it still wouldn't really affect much, since the
practicable teachings ascribed to "Buddhism" 1. answers questions to my satisfaction like no other religion/philosophy, and 2. those teachings are personally verifiable, to be "seen here & now, timeless, inviting verification, pertinent, to be experienced by the observant for themselves" (the teachings - the Dhamma - is frequently described in these terms in the Buddhist canon).
To borrow CARM's words, "Our confidence in the applicability and verifiability of the Buddha-Dhamma rests on a foundation that is simply without comparison."