Why doesn't electricty flow to earth ground when I connect my oscilloscope ground clip to a floating circuit?
The problem at hand might be relating the physical world to the ideal circuit world. 

There is no way for us to build this circuit (more on that later):



But if you could, there would be no charge flowing in or out of the ground connection.

In the real world we merely approximate things, we can connect a battery to a 1k resistor and connect a wire to ground, but the circuit really looks more like this:






simulate this circuit &ndash; Schematic created using CircuitLab

So there could be small amounts of current flowing to ground, the air can actually conduct a small amount of electrons, and there is a small amount of capacitance (and the whole circuit would function like an antenna of sorts.

We could even get more detailed than the model above! 

The real question is, how detailed to you want your circuit model? The more complicated the circuit model, the more accurate it is, but the more time and energy is need to solve it. 


  Because the clip is connected to ground with a low resistance path,
  and because the potentials are different, it makes zero sense to me
  whenever people say that no current will flow through there. If
  current is the movement of electrical charges, how is it physically
  possible that connecting a different potential would not result in
  charge movement?


In a model like the one shown above, there is no current flow. In the real world there could be a small amount of current.

As far as your capacitor circuit goes:


  I do understand that in the case of a small capacitor this effect
  might be small enough for us to ignore, as the capacitance of a small
  wire is probably beyond what we can even measure with test equipment?


The way this circuit is drawn, no current can flow through the capacitor, there is no other node for current to flow into. Both plates of the capacitor could be considered at the same voltage. Or the other node could be considered at an indeterminate voltage. (there is not a good way to calculate a differential voltage if you don't know the one of the values your differentiating). 

In the real world, this is not the case, A real capacitor looks like this:


The parallel resistance of the material between plates allows for a small amount of current that equalizes the voltage on the plates after a long time.