I noticed that there is a paradox in the episode The Big Bang, If the Doctor never existed, the human race would've died out long before Amy was born.
You're assuming that ---
Oh, Sexy is right! Tenses are difficult. The (natural) Big Bang I and the Doctor's Big Bang II both occur at the same time, from a simple temporal viewpoint, but there also exists a perspective from which II is "after" I. We'll need to adopt that perspective.
You're assuming that -- after Big Bang II "replaces" Big Bang I -- there is no Doctor in the history of the universe. If that were true, then none of the Doctor's adventures ever happened. He never landed at 76 Totters Lane because there was no original incarnation of the Doctor that could have stolen a sexy ol' Type 40. You're assuming that all of the disasters in Earth's history that the Doctor would have prevented, instead must have occurred disastrously. The world would be so different that Amy's parents might not have existed, let alone met. For example, without the Fourth Doctor,
you might assume that Scaroth the Jagaroth must have prevented the explosion that created life on Earth, so the human species cannot exist. (City of Death, 1979)
There exists at least one simple way to resolve this apparent paradox, and it is consistent with what we have already seen in this story arc. The Doctor is erased from history in much the same way that Amy's parents were removed from the Big Bang I universe. The young Amelia Pond of the Big Bang I universe quite literally has no parents. She exists without ever having been born.
Except, of course, she was born. She had parents. Her parents raised her for several years. Then, her parents were erased from history, but Amy was not. The Crack manages to erase her parents from history without affecting anything that her parents had accomplished. Their child still exists and this child still lives in the house that they bought, and Amy is still related to her Aunt.
You're thinking along the lines of something I call Butterfly Effect Time Travel. The Crack implements a different time travel model. It exhibits a Rubber Band Effect. When the Crack erases someone, history is changed, but only slightly, only minimally.
This is what happens after Big Bang II: First, the universe is restarted. All of history occurs in the way that it would have occurred without the Crack. The Doctor has all of the adventures that he had prior to the rebooting Bang, with the singular exception of his recent adventures that involved the now non-existent Crack. Then, the erasure effect from when Eleven pilots the Pandorica into the Heart of the Crack occurs. The Doctor himself is erased from history the same way that Amy's parents were erased. Everything the Doctor did in his personal history still happened, with essentially the same results, even though it all happens (in this iteration) in the Doctor's absence.
This is the Rubber Band Effect. History is changed, but it pulls itself back on track, running as close as it can to the earlier iteration of events.
After this, life on Earth survives every disaster that the Doctor prevented in an "earlier" iteration. It just happens to survive without the Doctor's presence. Amy is alive and well and about to get married. Even River is alive and well, despite the fact that
she was (originally) created solely for the purpose of silencing the (now non-existent) Doctor.
If you insist that the Butterfly Effect must apply to all time travel in Doctor Who, then you're gonna have a bad time. In the Doctor Who universe, some instances of time travel employ the Butterfly Effect, some employ the Rubber Band Effect, and some involve Tricked-Out Time. There are so many different methods of time travel and so many different conditions under which time travel happens. Different methods and different conditions produce different results.
With or without her parents, and with or without the Doctor, Amy exists. Amy is still affected by growing up next to the Crack, even in a later iteration in which the Crack didn't exist.
It's not a paradox. It's an acausal result -- a thing that still happens even though the "original" reason for its happening no longer exists -- that the Rubber Band Effect allows. Time isn't rewritten so much as it's carefully edited with a small bottle of white-out.
That's why Amy still exists, and that's why Amy can bring the Doctor back. All the original text is still there, and Amy's ability to remember means that she can scrape the white-out off the page.