Central tendency describes the point estimate with the highest likelihood of being observed should we repeat sampling with replacement from the same population.
Central tendency is of interest when the focus of analysis is to understand typical type of behaviour.
Robust analysis is insensitive (or less sensitive) to the presence of extreme values.
If the aim is to analyse typical expected behaviour and not understand the behaviour of extreme values, then there is no need to eliminate or even manage extreme values in any way. Instead, use robust methods of analysis.
For example, why estimate the arithmetic mean when one can estimate the median, or why estimate the conditional mean by minimizing the sum of squared errors using OLS regression, when one can estimate the conditional median by minimizing the sum of absolute errors?