Granted, this assumes that immortals of this type do not lose their memories over time. A recent Doctor Who character was a human who was inadvertently given immortality (or, at least, a very long life span), but still possessing a human mind, could not accurately remember the details of her past. To compensate, she wrote extensive journals of her life and experiences and would read through them from time to time.
This is like our history now. We have an imperfect written record of the events that preceded our existence. Think of all the events of our own lives that go by unrecorded. Consider how inaccurate the recordings of those few events we have written down are. We have no impartial observer with an eidetic memory to correct our misconceptions. Instead, we piece together a rough outline and try to fill the gaps with logical assumptions and our own past experiences.
Still, getting back to Connor MacLeod, I can't help but imagine the lives of people who could see our imperfect view of history and shake their heads at all of our false leads and misunderstandings. Would they try to guide the hands of those who write our histories to keep them more in line with what actually happened? Would they prefer to just go about their lives rather than waste their time in arguments that they had no real way of proving?
I feel like there's a novel idea in here somewhere.