Why are the kilogram, ampere, kelvin, and mole still considered base units and not derived? Given their current definitions, they're purely defined in terms of other units and cannot be independently measured, right?
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It's a multiplier needed to keep the new definition backwards compatible.
So why does this multiplier make it a base unit, but multipliers used for derived units do not? Does a unit become a base unit if the multiplier used in its definition is something other than a whole integer?
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They have different dimensions (time vs length) hence are different units. The choice of dimensions is a bit arbitrary so there can be a different set of base units. There are systems of so called natural units defined by setting various physical constants to (usually) 1.
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