Considering how old the Antarctic ice cover is, why isn't it much thicker?
Ice floats with gravity towards lower elevation, the flow continues until the base of the ice sheet becomes floating and the ice forms an ice shelf or icebergs. 

Due to the subglacial topography, basal melting and mass balance, the flow velocities vary over a large range, faster outflows are glaciers. The pattern is somehow similar to how rivers transport rainwater towards the coasts. 

The thickness of the ice sheet is controlled by a complex relation between accumulation, ablation and compaction.  


Flow velocities from MEaSUREs (NASA), plotted from Quantarctica

The ice velocity map shows, that some areas are unaffected by the flow towards the coast. Some of these areas are ice-free. Also at some inland locations, as Vostok, the velocity is very slow. Also, note that this is the surface velocity. At the base, the ice flows slower or not at all. 

To investigate the mass balance further, I recommended you to download Quantarctica and look closer at the datasets. Shepherd et al (2012) is a good first read about the methods used to estimate changes in ice mass and the present changes.