They preach that carbon dating is the only dating technique available.
It's not quite that bad. At least the example you cite here isn't far off.
Greyling is referring to methods of dating human remains and not for dating other things. Many of the other radiometric dating methods are totally inappropriate for that task. For example, you would generally not find an application for U-Th dating for most human remains. K40 dating is actually good for dates that are much greater than the ones discussed here.
Which radiometric dating means do you think Greyling overlooks?
What Greyling does not acknowledge here is indirect means for dating human remains. But I'd also note that admitting that radiometric dating is useful for dating humans is a real admission for a YEC.
Under a government which imprisons any unjustly, the true place for a just man is also in prison. Thoreau: Civil Disobedience (1846)
I believe that a scientist looking at nonscientific problems is just as dumb as the next guy.
Richard P. Feynman
If there is no struggle, there is no progress. Those who profess to favor freedom, and deprecate agitation, are men who want crops without plowing up the ground, they want rain without thunder and lightning. Frederick Douglass