It would have been easy to avoid the redefinitions of all other units by defining the second based on the frequency of an atomic clock that works in a null gravity field.

This would have required to apply a relativistic correction to the measured frequency of any atomic clock, but it would have provided unit definitions independent of the position in the Universe.

However, before contemplating the idea of time keeping on other celestial bodies it was decided to define the second based on an atomic clock that works on the surface of the geoid.

I believe that this was a big mistake, because it ties all the SI units to the Earth and especially because it does not really avoid the use of relativistic corrections. Now the precision of the atomic clocks is so great that for most of them it is necessary to apply relativistic corrections depending on the altitude of the laboratory.