Why is power so often associated with current and not also voltage?
It seems that whenever people talk about power, they are only really
  concerned with current and not voltage


Ignorant people perhaps, but every competent electrical engineer knows that both voltage and current must be taken into account.


  Although the small-signal voltage gain of the emitter follower is
  slightly less than 1, the small signal current gain is normally
  greater than 1.


There is no discrepancy here. The authors are pointing out that if there is current gain then voltage gain is not required to increase power. So the myth they are trying to dispel is that you can't have power gain without voltage gain - exactly the opposite of what you think people are concerned about. 


  One explanation I can think of is that there can be voltage across an
  open circuit, so increasing that voltage would theoretically increase
  power


If the output is open circuit then it will draw no current so there cannot be any power gain. However it may still be useful to consider the voltage gain if that voltage can be maintained with a load. 

In some circuits (eg. video amplifier) the source and load impedances are matched, resulting in half of the output voltage and power being lost in the source. In this case you would normally only consider the voltage gain under load (so a video amp with open circuit voltage gain of 2 is actually a unity gain buffer). 

In others (eg. audio amplifier) the load impedance is normally much higher than the source impedance, so the voltage gain remains (almost) constant whether driving the load or open circuit. If the load impedance is reduced then it will draw more current and power at the same voltage. This is not important for low level signals so usually only the voltage is considered. The output power of an audio power amplifier is very important, so speaker impedance is always considered - but output voltage and current are rarely mentioned.