Physicists tend not to like to have multiple conventions to cover differing but similar situations. They are much happier if they can come up with a convention that is more all-encompassing and covers multiple situations.
Take voltage for example. Instead of defining a voltage convention for positive charges and a separate one for negative charges, physicists prefer to define just one convention, and DEFINE voltage to be the electric potential energy experienced by a unit POSITIVE charge.
With this definition, a positive charge at a positive terminal has high POTENTIAL ENERGY (Energy = qV, where q and V are both positive), and a negative charge at a negative terminal also has high POTENTIAL ENERGY (again Energy = qV, where q and V are both negative). So one common definition of voltage (i.e. as experienced by positive charges) is enough to model the high ENERGY situation of both a positive charge at a positive terminal and a negative charge at a negative terminal. Note that the negative charge doesn't have high VOLTAGE at a negative terminal -- it has high ENERGY.
Another area in which the convention is based on the experience of the positive charge is current. Current is defined as the direction of flow of positive charge, even though we know that in a wire the charge carriers are electrons and negative. The physical electrons in a wire flow opposite the direction of current. Again, this is just an all-encompassing convention.
Regarding the electric potential of two point charges: the convention, as you point out, is to have the energy be zero when the charges are infinitely far apart. We could place the zero of energy wherever we want, but the existing convention is both convenient and more all-encompassing. With the energy zero for infinite separation distance, we can write just one equation for the energy of two point charges:
P.E. = q1 * q2 / R
This one equation encompasses everything we want to express about the energy of two point charges:
When the charges are close and both positive, P.E. is large and positive (very high potential energy).
When the charges are close and both negative, P.E. is also large and positive (very high potential energy).
When the charges are close and opposite sign, P.E. is large and negative (very low potential energy).
Note also that for charges with opposite sign and R -> 0, the energy approaches negative infinity. This would be a bad place to set the zero of energy (as you suggested in your question), since it is infinitely lower than the energy level than any other configuration of the charges, and so any other configuration would have to be listed as infinte energy by comparison.