The kilogram is not the only unit that will be redefined based on universal constants. The seven base units[0] will transition to being based on elementary charge and the Planck, Boltzmann, and Avogadro constants[1].
Four of the seven physical constants used to define the seven base SI units will change, so a bunch of physical constants that currently have exactly defined values will start being subject to measurement uncertainty. So μ_0 will no longer be exactly 4π × 10^-7 H/m, k_C = 1/4πε_0 will no longer be exactly 8,987,551,787.3681764 N m^2/C^2, and 1 mol of carbon-12 will no longer have a mass of exactly 12 g.
And for that matter, isn't it a bit frivol to call it a constant, if it's based on something mutable? Perhaps "the Avogadro number" would be a more appropriate name.
Slightly off-topic, but looking at the temperature section in your second link reminded me of the latest video from Linus Tech Tips[1] where he gets bashed for using the expression "degrees Kelvin". Personally, I don't see the problem. Since a kelvin is a hundredth of the difference between boiling and freezing temperatures of water, there is a notion of scale so the term "degrees" makes sense.