I wonder how much the public perception of the danger of large-scale ice melt is influenced by a lay understanding of the way phase changes can happen extremely suddenly.
Probably not that much. What worries me is feedback loops like the melting permafrost, and collapse of the Amazon ecosystem. Or the loss of ice causing more sunlight to be absorbed. Or a sudden overflow of the ocean's ability to store energy (something that has been masking most of the warming so far). Things could run out of our control in a very short amount of time.
I was talking about this to a fellow physicist friend the other day. Natural system with equilibrium points don't evolve away from them smoothly. It doesn't seem reasonable to expect the climate to be any different.