Question about the statement "Higher voltage means higher velocity of charge"
From what I understand about batteries, mobile electrons moving through the circuit can be derived from either the atoms of wire itself (i.e. external to the battery) or derived from the anode of the battery due to chemical reaction (i.e. internal to the battery).

That seems OK but be sure that you understand that the mobile charges are moving rather like a bicycle chain. When you stand on the pedal the effect is felt on the rear wheel sprocket &quot;at the speed of light&quot; while the chain itself moves &quot;at the drift velocity of the electrons&quot;. (Electron drift velocity will be in the order of mm/s.)

How do I know that as the voltage of a battery increases ...
A) more &quot;external-to-the-battery&quot; electrons are being ripped away from the atoms that comprise the wire and/or

They're not being ripped. A long chain of electrons is being nudged along by the battery.

B) more &quot;internal-to-the-battery&quot; electrons are being supplied to the wire?

The same number of mobile charges that leave on one terminal are returned on the other.

In either case, you would certainly have a &quot;greater total&quot; of electrons in the circuit as compared to lower voltages ...

No, there is no change to the number of electrons in the circuit.

... and therefore, assuming these extra electrons are mobile and moving at the same velocity as they were for lower voltages, the current would be higher as well.

You've lost me here. I don't understand what you are proposing.

So why do we know that this other possible phenomenon is not occurring.

By measuring the current?