Why is the Kelvin Temperature Scale Based on the Triple Point of Water?
In reality all the "primary" scientific values have changed over time as experimental techniques have improved. Science has evolved to the point where a set of seven "SI base units" (ampere, kelvin, second, meter, kilogram, candela, and mole) have been selected and defined so that better measurements won't cause disruptive changes in scientific data. 

So the gist is that the "temperature of ice water" depends on the atmospheric pressure. So it is necessary to define that 0 °C is the temperature of ice water at "standard atmospheric pressure." That means that if the "standard atmospheric pressure" changes, then so does the "temperature of ice water at standard pressure." 

However the triple point of water doesn't change and has a unique value. So by defining the triple point of water as exactly 0.01 °C in 1954, then the Celsius scale is anchored as much as the Kelvin scale. By also defining absolute zero as 0 K and −273.15 °C then the temperature scale itself won't change if better ways to measure temperature are devised. 

Thus by the time the whole notion of the primary base units was sorted out there was a massive amount of scientific data that had been accumulated using the old definition of the Celsius scale that was based on ice water at a standard atmosphere. Letting the triple point be exactly 0.01 °C kept all of that "old" temperature data "reasonably accurate" since 0.01° is only a 0.01/273.15=+0.0037% change.