Electric Potential concept
There are a number of assumptions circuit theory makes. For one thing it assumes that the potential along any point on a wire is the same if you don't pass over any circuit elements.
Secondly the concept of electric potential is a little irrelevant. We don't speak of electric potential, only potential difference for DC circuits. Remember that the electric potential is the amount of work done in bringing a unit of charge from infinity to a given point. But in circuit theory, we only speak of relative potential differences across circuit elements.
So a potential difference of 5V across a circuit elements could mean two things. One, it could mean that each Coulomb of charge gains 5J of energy as it passes through the circuit element. On the other hand, it could be that each unit charge loses 5J of energy when it passes through the circuit element. So you can see from here, that it's kind of true that bigger charges (more accurately called current since it's moving charges) dissipates more energy when it passes through a circuit element. Whether it gains or loses energy depends on whether it's a voltage source or voltage drain.
Circuit elements like resistors absorb energy when charges pass through them, so we always speak of a voltage drop across resistors. Batteries are voltage sources, so we speak of energy being given to the charges as it passes through the battery. So unlike resistors, when current enters from the positive terminal and exits the negative, current enters a battery from negative and exits through positive. That means the charges are being powered by the battery.
We don't speak of charges because we're not concerned about the electric field due to these charges unlike in electrostatics. Instead we talk about current, since ultimately moving charges is current. You should think of potential difference as the amount of energy supplied/depleted to/from each unit of charge as it passes through that circuit element per unit time.