It may be useful to understand where this error is coming from. Back when the theory of electrical current was developed, we had no good model of the atom. We had no idea about electrons, protons and neutrons, no idea about photons, quarks, nothing. Kirchoff developed his equations without knowing anything about electrons. When Ohm discovered his law, he knew nothing about electrons. Classical electromagnetism, in its entirety, is formulated without any knowledge of electrons (except with some of the late stuff that was developed in order to extend it to relativistic cases, but at that point we were already operating on "legacy code"). The modern idea of the atom, as we know it now, was developed more or less at the same time with classical electromagnetism; various theories about matter were being thrown around at the time, and for a long time, it was thought that electricity would be caused by some sort of charged fluid. Only beginning with the 1870s, at which time Maxwell was already revising his all-encompassing treaty, did the idea that it's all about particles really begin to take off.
The electrical charges in the classic theory of electromagnetism are not electrons, nor ions, nor any other particles that we know of. They're ideal models of "something" that carries charge, but they do not model all the inherent behaviour of electrons (e.g. the electrical charges of classical electromagnetism do not have any magnetic moments, unlike real-life electrons). They are somewhat like the point particles in kinematics, in that they capture an essential feature (electrical charge) of an object while dispensing with other features that are not essential for the study of some (but not all!) phenomenons.
It's ok to model electric current as "the flow of charge", as long as you don't give this model more physical meaning than it's due and attempt to equate the charges of classical electromagnetism with electrons.
Certain phenomenons can be studied within this frame: for instance, the motion of an electron's motion through static electrical and magnetic fields is generally (low enough frequencies, strong enough magnetic fields etc.) OK to do while equating the electron with the charge carrier of classical electromagnetism.
But back when the concept of electrical charge was elaborated, we knew nothing of electrons. The phenomenons gave enough clues for us to hypothesize that whatever "supports" electricity has some of the properties that "real" charge carriers have, but that's all there is to it.
The electrical charges in the classic theory of electromagnetism are not electrons, nor ions, nor any other particles that we know of. They're ideal models of "something" that carries charge, but they do not model all the inherent behaviour of electrons (e.g. the electrical charges of classical electromagnetism do not have any magnetic moments, unlike real-life electrons). They are somewhat like the point particles in kinematics, in that they capture an essential feature (electrical charge) of an object while dispensing with other features that are not essential for the study of some (but not all!) phenomenons.
It's ok to model electric current as "the flow of charge", as long as you don't give this model more physical meaning than it's due and attempt to equate the charges of classical electromagnetism with electrons.
Certain phenomenons can be studied within this frame: for instance, the motion of an electron's motion through static electrical and magnetic fields is generally (low enough frequencies, strong enough magnetic fields etc.) OK to do while equating the electron with the charge carrier of classical electromagnetism.
But back when the concept of electrical charge was elaborated, we knew nothing of electrons. The phenomenons gave enough clues for us to hypothesize that whatever "supports" electricity has some of the properties that "real" charge carriers have, but that's all there is to it.