Why is the ampere a base unit and not the coulomb?

Because it was defined by measurements (the force between two wire segments) that could be easily made in the laboratory at the time. The phrase is "operational definition", and it is the cause of many (most? all?) of the seemingly weird decision about fundamental units.

It is why we define the second and the speed of light but derive the meter these days.

Since this question was asked, the situation has changed: there is movement towards a redefinition of the SI system which eliminates arbitrary artifacts in terms of quantities which quantum mechanics tells us are really, fundamentally constant. Starting sometime in 2018, the defined constants will be

  • the difference in frequency $\Delta\nu$ between two particular electronic transitions in cesium atoms (unless a more stable technology is developed)

  • a constant $K_\mathrm{cd}$ defining the candela

  • the speed $c \approx 3.0\times10^8\,\rm m/s$ of light in a vacuum, relating distance to time

  • the quantum of electric charge $e \approx 1.60\times10^{-19}\rm \,C$

  • the Planck constant $h \approx 6.6\times10^{-34} \rm\,J\,s$ relating the charge quantum to the magnetic flux quantum, and also relating wavelength, momentum, and mass

  • the Avogadro constant $N_A \approx 6.0\times10^{23}\,\rm mol^{-1}$ relating the kilogram and the atomic mass unit

  • the Boltzmann constant $k \approx 1.38\times10^{-23} \rm\, J/K$ relating temperature and thermal energy.

In the present version of SI, the first of these three are exactly defined, while the other four are empirically measured based on the international prototype kilogram, the magnetic force measurement used to define the ampere, the mass of a mole of carbon-12, and the triple point of water. All of these are macroscopic phenomena. After the 2018 redefinition all seven of the constants I listed will be "exact" in the way that $c$ is exact at present.

There's more information about the SI overhaul at the BIPM, on Wikipedia, and at NIST. Here's also a Nature news story about the redefinition of the ampere.