> in the physics that is encountered by the majority of people there are only 2 fundamental constants, which determine the strength of the electromagnetic interaction and the strength of the gravitational interaction, i.e. the constant of Coulomb's law of electric attraction and the Newtonian constant of gravitation

The speed of light in a vacuum is pretty important too, but I don't think that you can derive that exclusively from the Coulomb constant (without also allowing the permittivity/permeability of free space). Planck's constant is also important, depending on what you consider "physics that is encountered by the majority of people".

> The poor handling of metrology in modern physics textbooks

It's not really covered in textbooks, but all my physics labs had extensive coverage of error propagation, dimensional analysis, and running statistics on measured data.

> In the SI system of units, besides the 2 constants of Newton and Coulomb there are a few other "fundamental constants", which are fake fundamental constants

I agree that Boltzmann's and Avogadro's constants are "fake", the elementary charge is borderline, and the candela is a joke, but I wouldn't put c or ℏ in this category—time is special in physics (the arrow of time and such), so the relation between time and space (c) seems fundamental and non-trivial; similarly, conservation of energy is probably the most important conservation law, so the relation between frequency and energy (ℏ) seems fundamental too.

I'm aware of natural units, but I don't think that the fact that you can give c and ℏ values of one makes them any less fundamental.