But even then the equilibrium constants are constant; we just can't approximate with concentrations 
You are very correct. I attempted two or three times to write something about counter-cases, but they are very nit-picky and not very useful, so I deleted them.

The point is though, equilibrium constants are taught and used because it is a simple tool that is surprisingly accurate in its predictions. When its use become so complex that it is no longer a simple tool, it becomes a rather irrelevant concept. E.g. we wouldn't calculate cell voltage in batteries via the equilibrium constant, because calculating/measuring activity coefficients in a MnO2 sludge is difficult. Instead, we'll simply build the battery and test its voltage.
Which leads me to ask a question of my own.
In the VCE course, we are told that brine, a 5 M solution of sodium chloride, will electrolyse to form sodium hydroxide and chlorine gas. Now, the Nernst equation in its standard form doesn't predict this, so I'm presuming the discrepancy arises from the 5 M concentration and the non-ideality of the solution?
Yes, but also importantly, the Nernst equation describes the
equilibrium cell potential when the solution is static (i.e. reaction rate is zero). While the cell is not
in equilibrium static (which, in this case it is not), the Nernst equation does not directly apply.
Note for the general readers: if you don't quite understand what I mean, don't worry. This isn't even 1st year university material. Only people who specialise in surface chemistry and diffusion processes in research eventually learn about methods to fix this problem. That makes this... at least Master's or PhD level stuff.