If our system was based on Planck units then it would be interesting. It would also cause tons of other fundamental constants to be greatly simplified to either integers or integer multiples of known transcendental constants.
The so called Planck units are the worst system of units conceivable and they could never be used in practice. This has nothing to do with the values of the Planck units, but with their uncertainties.
When Planck has suggested that system of units, as a possible improvement over the system of natural units proposed by Maxwell a quarter of century before him, by removing 2 somewhat arbitrary choices required by the Maxwell system of units (of 2 kinds of atoms, one for providing a frequency unit and one for providing a mass unit), that was before the development of quantum mechanics and before of the discovery of several quantum effects that are useful in metrology.
The reason why the Planck system of units is bad is because it defines the Newtonian constant of gravitation as an exact constant.
However, the Newtonian constant of gravitation can be measured only with an extreme uncertainty, many, many orders of magnitude greater than the uncertainty for measuring any other fundamental physical quantity.
By forcing the Newtonian constant of gravitation to be exact, its uncertainty does not disappear. That uncertainty just moves into the values of all other physical quantities that include mass in their dimensional formulae.
This means that in Planck's system of units most absolute values of physical quantities have uncertainties far too great to be usable. In Planck's system of units, for most quantities only the ratio between 2 quantities can be accurate, not also their absolute values.
Nevertheless, not all is bad in Planck's system of units. Only using the Newtonian constant of gravity is bad. Using the Planck constant to provide a unit of mass instead of using the mass of some arbitrary atom is good.
By combining Maxwell's system of units with the good part of Planck's system of units, you can obtain a system of natural units where there is only one arbitrary choice, of an atomic transition that can provide a unit of frequency. All the other "fundamental constants" can be defined as 1, with the exception of 2 constants that must be measured experimentally, and which provide the intensity of the gravitational interaction, i.e. the Newtonian constant of gravitation, and the intensity of the electromagnetic interaction, i.e. the so-called constant of the fine structure, a.k.a. Sommerfeld constant.
After its last revision, the International System of Units has actually become equivalent with such a Maxwell-Planck system of natural units, except that this is masked for historical reasons by the use of a large number of "fundamental constants" that are inserted into the relationships between physical quantities, and which are exact, but instead of being equal to 1 they have various weird values.
For theoretical work, or inside some simulation programs, it can be more convenient to use a system of units where all "fundamental constants" are 1, and where the unit of time is taken to be the period of the electromagnetic wave corresponding to the cesium 133 transition on which the SI is based (i.e. about 0.109 nanoseconds), so that any value in such a system of units can be converted by an exact factor into a value in SI, e.g. for displaying the results. (Actually that is what I always do.)
