Of course they think carbon dating is accurate. Consider the source. In reality, they have no way of knowing; I certainly do not presume to know either, but they doassume, and they do call it fact. We can't see much of the decay in the half-life of carbon, but what we do see is linear, so they assume that it stays linear. That is something I learned in science class that I actually appreciated.
And just because the ratio of a million in 4.5 billion is similar to one in 100 does not mean they are the same. Even one million years is really more than anyone can fathom. I mean can you even wrap your head around that figure? 4.5 billion years? Or 14 billion years for the supposed age of the universe? No way can a human being understand that, but we see just see 4.5 and 14 rather than 4,500,000,000 and 14,000,000,000. Then it's easy to say, "what's a million years difference? It's all been around for billions of years anyway." The fact still remains that if scientists can't tell if a rock is 570 million years old or 568 million years old, that is a really big difference, and something is wrong. 570,000,000 and 568,000,000 are very different--2,000,000 off, and I'm sure there have been far larger discrepancies.
Something else to take into account: the carbon decay that they find could be from just about anything else in the surrounding area. Carbon is everywhere, and that is a blessing and a curse for scientists. That means they can use it to date anything, but it also means that the things they date could be very innacurate due to everything else around with decaying carbon.
Lastly, time--as in 1:30 PM, 10:45 AM, et cetera--is a figment of the human imagination. It was brought about during the industrial revolution in the mid-sixteenth century. But time--as in one day and one week--was a product of God. Genesis 1:4-5 : "4 And God saw the light, that it was good: and God divided the light from the darkness. 5 And God called the light Day, and the darkness he called Night. And the evening and the morning were the first day."
God decided what a day was, and perhaps it wasn't 24 hours then due to the earth spinning faster or whatever it may have been, but it was still sunrise to sunset.