I'm going to open with a quick little metaphor. Fable. Something.
Two people are sitting down to dinner. They've agreed that they'll both have the same dish, but they have two choices to pick from (without getting a taste test of either first). One looks so-so but smells amazing; the other looks amazing but smells only so-so. Unfortunately for our diners, one of them is blind and the other is an anosmic; both have been afflicted since birth. How can they come to an agreement over what to eat? The most salient evidence available to one party is practically nonsense to the other.
If you cruise the atheist/science Internet, you will come across quite a few stories of conservative religious people who came to science as a result of scientific evidence and debate, and all the more power to them. However, when you consider the kind of traction that religion, Christianity, and young Earth creationism still has in American society, you have to admit that this method isn't accomplishing much: those people are the exception that proves the rule. Scientists are a bunch of anosmics trying to explain to blind people why the dish with the more appetizing presentation is the way to go (or a bunch of blind people trying to describe just how mouth-watering the other dish smells, take your pick).
Fossil records and carbon dating do not and will not (in most cases) convince the typical Young Earther to change their mind, in the same way that accounts of miracles or God's providence will not (in most cases) convince scientists. They're like visual accounts to the blind, or descriptions of smell to the anosmic. The two different groups have such different ways of prioritizing and sorting information that they forget: a case that convinces themselves and their peers is NOT going to convince the people they're trying to convince.
Why do Young Earthers try so hard to discredit things like fossil records and carbon dating? Because they get that if they can somehow discredit the science of anything, that the scientists will admit defeat. They've realized that they've got to talk the science language and play the science game to even have a chance at "winning" the debate. In that sense, they're one step ahead of the other side, because no one on the science side is as public and strident about playing the faith game.
If nonsense non-science is ever going to lose traction in the public's view, it has to be because not believing in the non-science won't carry the stigma of being a bad Christian, NOT because the science itself is airtight. Being religious is fundamental to an identity in a way that being a scientist often isn't. For many people being a good Christian means they have to reject evolution (because the loudest leaders at the top of the faith pyramid say they have to), and nine times out of ten the identity they have as a "good Christian" and the desire they have to remain part of one the more important social groups in their lives is more salient to them than good science.
Debates like the one between Bill Nye and Ken Ham will always be counterproductive for that reason. What we need are debates between Christians and Christians that avoid the science entirely but instead tackle the socio-theological issues that are at the root of this (i.e. Biblical literalism, the top-down nature of some denominations of Christianity). When you can change what it means to be a "good Christian" by the group's standards, that's when progress can begin.
For a helpful model, refer to the debate between Christianity and gay marriage. As often as people have been arguing that some people's religious beliefs shouldn't be the basis for national legislation (that is, making an argument that disregards the primacy of faith for many believers), people have also been using the Bible itself to defend the thesis that gay rights and Christianity are not incompatible. Matthew Vines is an example that immediately springs to mind.
Wake me up for that debate. But $favorite_famous_scientist and $loudmouth_evangelist? No thanks, I'll pass.