I think that Planck units are the most unpolitical of them all
Somehwat simplified - the universe has one scale which you need to pick to assign units to everything. It can be a length scale, a time duration,... it doesn't really matter which, it just has to be a dimensionful quantity.
The rest falls then naturally into powers of that unit if you refuse to add extra units.
Particle physicists like to pick the GeV as basic unit for energy. Then the uncertainty relation says that the conjugate to energy - time - is 1/GeV. Then General Relativity says that space and time are really the same, so length is also 1/GeV. The conjugate of length is momentum, which then gets to be GeV like energy (would also directly follow from GR).
Velocity is then length/ time and comes out dimensionless - if you're moving the same length and time, Special Relativity tells you that you're moving at lightspeed, so c is dimensionless and just gets the number 1, every other velocity is a number <1 and measures velocity as fraction of lightspeed.
Area is then 1/GeV^2 - volume 1/GeV^3, energy density is energy.volume, i.e. GeV^4,... and so on.
You could have picked meters as length units, or lightyears, or parsec, or the mean distance of the hydrogen atoms in a water molecule - say a unit U - then time would also be measured in U, energy and momentum in 1/U, and lightspeed would still be 1.
Which is universal - in natural units, whatever you pick, lightspeed is just unity because space and time is really the same.