Using an altitude filter, the label of the aircraft can be suppressed by setting a lower limit just above aerodrome elevation.
But in our case the system will throw a 0 on the mode C readout if it doesn't work for any reason, and that will then be corrected to a minus X amount of feet if the QNH is low..... so we never use the lower limit of a filtering function when working approach since we may loose the label if an aircraft has a mode C failure.
But for us with an "old" secondary surveillance radar having labels all over the airport can be slightly annoying for approach. But it all depends on the radar system... and I bet there are quite a few out there more advanced than ours