The claim is that these plants haven’t been exposed for thousands of years, as dated by the C14 isotope. At four different ice caps, radiocarbon dates show the  mosses had not been exposed to the elements since at least 44,000 to  51,000 years ago. That might be true, but then again they are long dead, so there  wouldn’t be any uptake of new C14 if they were exposed to the open air  in the past. There’s no claim that the mosses are now suddenly alive and  growing again. So, if they had been “exposed to the elements” since  then, they would not have an new C14 in them unless they came back to  life and conducted photosynthesis.
 Since plant material in the Arctic doesn’t decay like it does  elsewhere due to low temperature and low humidity, it could very well  remain intact while exposed for quite some time. All I think they can  claim is that the plants haven’t been alive for 44,000 to 120,000 years.  I don’t think they can’t prove with C14 dating that they have not been  exposed then reburied under ice/snow since then. Ice is a funny thing,  it can melt due to warmer temperatures or it can sublimate at below  freezing temperatures if there’s not enough sustaining precipitation, as  we 
know from Mount Kilimanjaro.  What I’d really like to see is what the receding ice edge looks like.  Sublimation leaves a signature that is quite different from melting.