The charging current warms the battery as it charges, and because of mass-energy equivalence, an unmeasurably small amount of mass is indeed added to the phone in the process. However, this is not what is measured by your electric utility in order to bill you for energy.
The insignificant mass added to your phone when charging is lost partially when the heat from charging the battery radiates away, and also is diminished whenever the phone is actually in use, in either transmit or receive mode.
On the way to your home phone charger from the power station, a very small amount of energy of the electrons is also lost by heating transmission lines and transformers as the energy they carry is transferred.
The conventional direction of current flow is just that: a convention. It doesn't mean that the positive charges in the wires need to have a net flow into your home, or even that the same electrons that left the power station need to be deposited in the wires at your home in order for an electric current to flow. All that is really necessary is for the energy of the electrons at the power station to push the electrons in the wires in the direction of the electrical consumer's house, and as this is done with alternating current (50 or 60 Hz), no net deposition of mass, particularly electrons, needs to happen anywhere in the process.