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It's not a mistake, and conventional current is not wrong or backwards.

Electric current is often thought to be a flow of electrons, but this is wrong. Electric current is a flow of electric charge. Charge can be positive (protons) or negative (electrons), and both types of charged particles can and do flow in electric circuits:

(The Hall Effect can be used to show whether a charge carrier is positively charged and flowing in one direction, or negatively charged and flowing in the other.)

When a Lithium-ion battery discharges into a load, for instance, there is no electron flow in the battery, but there is still a current flow:

Lithium-ion battery discharging into resistor

(Source: Center for Sustainable Nanotechnology)

If you considered only the electron flow, your calculations would be wrong. You need to consider the net flow of charge, no matter what the charge carriers. Conventional current abstracts away the different charge carriers and represents all of these different flows as a net flow of (positive) charge, simplifying circuit analysis.

Conventional current is not the opposite of electron current, so if they were defined to flow in the same direction, it would be even easier to confuse them and go through life misunderstanding what current is. Electron current is a subset of conventional current. Conventional current combines the effects of electron, ion, proton, and hole flows all into one number.

Wikipedia agrees:

In other media, any stream of charged objects may constitute an electric current. To provide a definition of current that is independent of the type of charge carriers flowing, conventional current is defined to flow in the same direction as positive charges.

The labeling of one polarity of charge as "positive" and the other as "negative" is totally arbitrary. It could be done either way and everything would still work out the same. Franklin didn't choose wrong; he just chose. Labeling protons as negative and electrons as positive wouldn't change anything. It might actually make things more confusing, as described in Ben Franklin should have said electrons are positive? Wrong.

If Franklin had instead chosen the electrons to be positive, then we might never confront the real problem. If electrons weren't negative, we'd easily ignore our misconceptions, and we'd end up with only an illusion of understanding. Yet also we'd still have all sorts of niggling unanswered questions caused by the misconceptions. Fortunately the negative electrons rub our noses in the problem, making our questions grow into something far more than just "niggling!"