Coulombs law as well as Amperes law and similar mathematical formulations of two centuries ago, were incorporated within the strict mathematical format of Maxwell's equations .
The apparently disparate laws and phenomena of electricity and magnetism were integrated by James Clerk Maxwell, who published an early form of the equations, which modify Ampère's circuital law by introducing a displacement current term. He showed that these equations imply that light propagates as electromagnetic waves.
Coulomb's law can be derived from the first of Maxwell's equations in this list.
Solutions of Maxwell's equations are what we are using to communicate on the net with, let alone all the electricity usage , wireless etc. Our technology rests on Maxwell's equations being valid.
One does not need to do long range experiments with individual charges because the law will hold for all distances where Maxwell's equations are valid. This means non General Relativity range where it has to be adapted. A version of Maxwell's equations exists in the quantized theories too .
If the first law from which Coulomb's equation is derivable, were not valid over the whole distances (all of earth, and laser light to the moon) where Maxwell's equations have been fundamental in constructing all our technology, there would have been discrepancies and Maxwell's equations would have been invalidated.
Here is a relatively recent (1970) test of Coulomb's law which gives an accuracy of $1.3 \times 10^{-13}$, compared to Coulomb's measurement (two hundred years before) of $4 \times 10^{-2}$. Concentric spheres are used and dimensions are of order of a meter.

Note that the technique relies completely on Maxwell equations being exact, as it uses electromagnetic waves for the detection of an anomaly.