rate doesn't affect the fact that accumulation is happening, what matters is that addition exceeds removal over time. How long it would take to make a thick pile, though, that does matter on rate, or more particularly the size of the difference between addition and removal (high addition rates combined with high removal rates would still result in slow accumulation, same as if both were low). The Antarctic has been accumulating ice for maybe as much as 25 million years, or about there, and it probably wasn't such a desert for all that time.

The earth did not go from "Antarctica is a temperate forest and grassland" to "Antarctica is a frozen desert of ice" overnight.