Importance of zinc blende and wurzite crystal structures for semiconductors?
As suggested I can try to summarize the basic idea in an answer although I wouldn't say it solves the bounty. I'm also unsure why you even set the bounty for it. 

For this I'll quote the book that I found a few days ago. As I own the missing pages now as well the answer seems a bit more elaborate then before. As the book is in German I will translate the text. 


  The formation of the band-structure of typical element semi-conductors like diamond (C), Si and Ge was already shown in chapter 7.3. By mixing s- and p-wavefunctions we'll get a tetrahedral bonding orbital ($sp^3$), that will lead to a splitting into a bonding and an anti-bonding orbital in the range of the equilibrium-bonding distance. The bonding orbitals form the valence band and the anti-bonding orbitals form the conducting band. The distribution of the four s- and p-electrons into bonding and anti-bonding orbitals leads to a completely filled valence band while the conducting band remains empty. 
  The result is therefore an insulator like diamond or, if the energy gap between the valence and conducting band is smaller, to semi-conductors like Si or Ge.  


I'll skip the next pages that deal with the formation of the band-structure and the symmetry aspects as I have no experience in these topics and continue with the $III-V$-semi-conductors. 


  As the formation of $sp^3$ hybrid orbitals in the chemical bonding of Si and Ge seem to be the a substancial part of the semi-conducting properties it seems reasonable that other materials with a tetrahedral crystal structure, i.e. $sp^3$ hybridisation, should show semi-conducting properties as well. 


At this point I will switch to basic solid state chemistry as the book doesn't elaborate this part well enough. As solid state chemist I often have examples like these with my students. A while ago we were discussing the silver-mobility in high-temperature $AgI$, which shows quite some conductitiy through defects and the low temperature modification. At this point I would typically ask for reasons why $AgI$ might be problematic. Typical answer that I expect are:


HSAB-concept  with soft $Ag^+$ and soft $I^-$
A covalent three-dimensional tetrahedral network with ZnS-structure


For the second part they either check these facts before we talk about it or they can use the things they have learned in lectures. One of them being the Grimm-Sommerfeld rule. Basically, compounds, that form between two elements to the left and to the right of another element will likely adopt the structure of said element. So for a group $IV$ element like the silicon structure, it can be $III-V$ or $II-VI$ or $I-VII$. It's one of these rules that are taught in the very first semester but people forget about it although it's super useful for compounds you don't know. 

Therefore my idea would be here that if, as the book mentions, $sp^3$ hybridisations and covalent, tetrahedral structures as the main source for the semi-conducting properties it's only natural that $III-V$ compounds show a similar effect. Those bands are basically molecular orbitals. If you start from a di-atomic molecule and increase the amount of bonding partners you will get more and more orbitals that move closer and closer together until you have bands. If let's say Ge has a certain band-gap and you change to GaAs, or course the single contributions to the HOMO and LUMO and later to the bands will be different but not that far away from Ge, so we expect a comparable band-gap.

Source: H. Ibach, H. Lüth, Festkörperphysik: Einführung in die Grundlagen, 7th edition, Springer, 2008.