Transistors: Switching a higher voltage at the source (not the drain) side
Is this PNP/NPN pairing a sensible way of doing it


Yes.


  Note that both of the transistor pairs share the same emitter resistor
  (since only one is enabled at any one time). I suspect that this
  prevents the NPN from getting to saturation since that resistor would
  pull the NPN's emitter voltage up above 5V.


Correct


  My solution for this...


No 'solution' is necessary, because it isn't a problem.


  Is is just a matter of setting that resistor to a lower value so that
  the PNP can, at least, saturate?


If 4.4mA isn't enough to put the PNP transistor into saturation then a lower resistor value will be necessary.