Why Coulomb's Law is only valid for point charges?

If you had, say, an extended object with many point charges and wanted to know the force on a point charge, Q, you would use Couldomb's Law to add up the forces on Q due to each individual charge. In the case of a continuous distribution of charge, you do a similar thing but consider the object to be made up of an infinite number of infinitesimal charges, dq, i.e. you do an integral. So in that sense Coulomb's Law is used for finding the force of the extended object on the charge Q.

But in general you can't just take the total charge of an arbitrarily shaped extended source with an arbitrary charge distribution and use Coulomb's Law to find the force on an external point charge (an exception is a spherical distribution of charge). One issue you would face is what do you use for the distance between the external point charge and the extended source?