The answer is in your question: "It was hard for me to believe that ice, like water, seeks its own level to that accuracy." - Not intuitive perhaps, but over time, ice will indeed creep and flow in response to stress.
Because the density of ice is larger than air, the weight of the ice makes a larger contribution to the hydrostatic pressure in the Antarctic ice cap (glacier) than does the air pressure at the surface. The layer of ice covering Antarctica is very thick, and the ice is a solid near its melting temperature. The ice creeps (flows) and the thick ice layer moves so as to become closer to an isostatic equilibrium state (lower energy.) Of course Alvarez would have been aware of this explanation.
The glaciers of Antartica also flow from the interior to the edge of the continent. The edges of the glaciers are wasted into the sea and snow falling in the interior provides a source of ice to the glaciers.
Thanks for mentioning this book. I think it's an interesting quote, because I expect it was made at a time (1960's?) when the high-temperature creep of geological materials (rocks and minerals) were an active area of experimental research at the University of California. It would be interesting to know what prompted the comment, and what discussion on this subject he may have had with David T. Griggs and his students at UCLA.
I believe he accepted the explanation as the correct one, but I would like to know whether he withheld that until there was laboratory evidence of ice creep.