Historically, it's half-European as well as half-Middle Eastern. And as for confessionally, culturally and politically, Atatürk did make it rather European in many ways even if it's predominantly Muslim, which in any case is also true for Albania (and Kosovo) and parts of Bosnia.

You may have missed the part where the EU has a long list of requirements that candidate members have to fulfill before they're allowed to join (incidentally, candidate members are not invited, they apply themselves), or the part where Turkey has made rather significant progress over the past ten years thanks largely to those entry requirements. Of course there's still a way to go, but the progress they have made on all of the issues you name - Armenia, the Kurds, the exaggerated power of the army, religious freedom for minorities - as well as general economic progress is impressive.