I think probably electric current, in its simplest definition, addresses the flow of electrons from a high electrical potential in a circuit to a low potential. It is probably treated as a fundamental quantitiy based on the principle of electron flow through the conducting material. Current in and of itself can be made to change, as you have pointed out, and thus be variations on a theme. Electric current through a defined material, at a certain temperature, with a specified length, will give consistent results each and every time you try it, which would tend to lend itself towards a fundamental definition. Electricity itself, has a specified rate at which it travels through individual conducting materials, so maybe that is why current is spoken of as a fundamental quantity. Electricity has been experimented with and known about since antiquity, so maybe this is a carryover from preexistent knowledge and application.