I read the link, but I don't see a definition of destabilization. The closest thing was rising sea levels would unanchor the ice sheets and cause them to break up. That is far flung as you know, sea levels would need to rise orders of magnitude more than current trends and predictions for that to happen. Another candidate is the breakup and darkening of the ice surface from melting. But that clearly isn't happening except on the fringes. Most of it is whitening, just the opposite. "Destabilization" seems to be qualitative and speculative.
The article points that out. "Destabilization" (my term) refers to what the article describes as increased melting leading to accelerated disintegration; major sea-level rise isn't necessary to initiate large-scale disintegration. Note that the article points out that the process is expected to be non-linear (meaning hard to predict), and that once "large-scale" break up starts it will be virtually impossible to stop.
A lot of ice sheets are prevented from faster flow to the ocean by the ice tongue or outflow area that is anchored to the bottom. If this area becomes disconnected from the bottom, ice sheet flows can accelerate.