Why is it that when calculating the induced emf in a loop of wire, we use flux?
Because of the Faraday law. It is a physical law that was obtained as generalization of experiments. There wasn't a way to derive it mathematically or logically from some pre-existing knowledge when Faraday discovered it.
The area that's used for the flux is the area of the plane of the loop, but isn't that space in many cases just air?
Yes, but the details of what is in that space do not change the Faraday law; the Faraday law states that the only thing that matters is the magnetic flux. The details of what is there will change the flux, but won't change the law.
It's not the conducting material covering that entire area
An induced vortex-like electric field is present whenever magnetic field changes. Consequently, emf assigned to a static closed loop which is in this space will have nonzero emf. This does not require presence of any conducting material.
We talk about induced emf in the wires because there it is practically interesting - the induced electric field in the wires or conductors can affect electric current and this effect is quantified via emf.
For example, in coil voltage transformer, the induced electric field is inside and in the vicinity of wires but also further away from them, in the core. But in ideal case the effect of this electric field in the wires is far more important on the operation of the transformer - it is what influences the electric voltage on the terminals of its secondary coil.
Also, in the text book we use there is a given example that's used to explain why relative motion between a conducting straight wire and a magnetic field produces emf. ... but there isn't any change in flux going on here. The wire is moving through the field, but it's uniform so the field on the wire isnt changing , and niether is the area or angle between them. So wouldn't an emf NOT be induced in this case according to faraday's law?
It helps to make clear what we mean by emf for a piece of path in space. It is work that would be done by electromotive agent (chemical or electric force) if one elementary charge was transported through that path.
In the case of isolated wire in motion through magnetic field, initially due to magnetic field charges would move along the wire and the moving wire would do some work on them. But very soon enough electric charge would build up on the ends and surface of the wire and counteract further motion of charges with respect to the wire. The agent moving the charges is still present (wire moving in the magnetic field) but it completely counteracted by equally strong force of electrostatic repulsion.
One way to describe this is that emf due to motion in magnetic field is counteracted by the emf due to electric force of accumulated charges. So, there are actually two different emfs, of equal strength but opposite sign. The total emf is zero.
The flux in this example is not a meaningful concept because there is no closed path indicated. Therefore this isolated wire in motion is not a good example of the Faraday law. It is better thought of as an explanation of where the force moving the charge carriers comes from and how the Faraday law for closed circuits moving in magnetic field can be understood as a result of magnetic force acting on the moving wire.
