Short circuit = no power?

You should not be so hard on your professor.

Much of the confusion newcomers to EE struggle with is that we talk about theoretical IDEAL circuits as part of the teaching process. In ideal circuits things often act rather contrary to your intuitive and experimental notions of how things actually work.

Things like short circuits, transformers, diodes, and pretty much everything else we work with, have ideal models we use to describe and understand them within the scope of how we try to use them. The reality is far more complicated and much harder, if not impossible, to define entirely.

As such the definition of a "short circuit" is in fact an "ideal component". It is a resistance with zero resistance, that is \$0\Omega\$. That is, the force of the battery will act through it with no opposing force. Pushing on nothing, you do no work, and no power is dissipated.

In real life of course, the wire you use to short out the battery has some small resistance. The battery itself also has some internal resistance. Since both of those are small, the resultant current is very large. That means lots of power is dissipated in the wire, and in the battery and things quickly get rather warm.

As I said, do not be so hard on your professor. A lot of EE is accepting the ideals at face value while realizing that reality is rather different. The ideal models give us a base point to work from which allow us to design things to a working level of accuracy without getting lost in the chaos of real world effects.

However, we always have to be mindful that the ideals are a myth.

even touching both ends of an AAA battery with a metal wire produces sparkles and heat

To analyze this circuit, you have to consider both the internal resistance of the battery and the actual resistance of the wire.

Since a real wire has non-zero resistance, some power will indeed be delivered to the wire and turned into heat.

But also, since a real battery has internal resistance, some power will be converted to heat inside the battery where it doesn't do any good and may damage the battery.

The statement (from the website) is correct only in a purely theoretical sense, as there really is no such thing as a 0 ohm short. All wires have some resistance, and a battery itself has internal resistance. Your professor was indeed correct - if there is current flow, then there is a voltage drop, though it may be very small.

In fact, one way of measuring current in a circuit is to place a small calibrated resistance (called a shunt resistor) of typically 0.01 ohm in series with the load, and measuring the voltage drop (usually in millivolts) of the shunt.