The only real danger here is that the EU is a much weaker project than assumed, not attractive enough to be able to generate significant change in prospective members. If so, it's doomed anyway.
Otherwise we assume it is able to do that. In which case a Turkey in compliance with specified EU standards for democracy and human rights is an asset.
EU has only been able to stimulate and sustain positive change in
European countries. Please accept that as a fact. The reason why countries like mine were so eager to accept pretty much everything the EU had requested was the notion that we were "returning to Europe" - "návrat do Evropy" as it was called. Whatever idiots like Klaus claim, Czechs (and Poles, and Estonians, and Slovenians, etc.) consider themselves as Europeans and they've always seen the EU membership as a proof of that. A symbol that the long separation from the European mainstream that had been forced on them by outside powers was over, gone, never to return.
In short, the EU was so successful in helping to reform Central Europe because the Central Europeans themselves were willing, even eager to reform. That sort of sentiment is largely or entirely missing in countries that lie outside the European cultural sphere.
The EU has now reached almost its final territorial extent. There aren't many countries left that share this ethos, this will to be a part of the great European project. There is the Balkans, sure - Croatia will join soon, followed perhaps by Serbia, Montenegro, and in the long run Macedonia, Bosnia and Albania. All these countries and their people see EU membership as their great aspiration. Who else is there? Well, Eastern Europe and the Caucasus, and the leftovers in Western Europe. The latter can join whenever they finally decide that it is what they want. Eastern Europeans have to settle the long-standing dispute between pro-Western and pro-Russian/Asian tendencies. In the Caucasus, Georgia made it clear it sees its future in Europe, rather than in Asia or Russia. Armenia is complicated, Azerbaijan is even more complicated. Israel is an outside possibility.
And that's it. Turkey has never felt the same about Europe. It applied for membership during the Cold War not because of any great love for Europe, but because it wanted to associate more closely with the West in order to increase its security. That motivation is now gone. EU membership is now strongly supported there only by a minority of pro-Western elites, while the rest simply doesn't feel European and if they want to join, it's purely for the economic benefits and free travel opportunities. In this Turkey is not different from Morocco, Tunisia or other countries along the Mediterranean, the only difference is that Turkey is for some reason treated differently, probably due to the Cold War inertia.
The EU cannot remake countries, to think otherwise would be even more arrogant than the American belief they could bring liberal democracy to Baghdad in their army trucks and helicopters. We cannot reshape Turkey into a Western, European nation by offering economic and political incentives. It is not in our power, so why don't we finally face that fact? Why don't you?
And it is an asset because, provided it works, all these dinky Central Asian republics will be more likely to look to Ankara to work out what they want to be like. If Ankara is firmly in the EU orbit, that's the direction they will be likely to want to head. Turkey is potentially the vehicle for the EU political and social ideals into the region. No EU Turkey, and they might as well look to Moscow...
So, you think the EU should take Turkey in to... get itself involved in the Central Asian mess? Wait, was that supposed to convince me it was a good idea?
And of course I have to argument against your specific notions about geography and Europeanness. Except that they're just arbitrary. Their only really stable aspect is that they deny universality. Which otoh tends to be the central to the principles on which the EU assumes to foot it's political and social ideals, meaning geography has no bearing on them.
It's not geography, it's culture and mentality, and you cannot handwave these away. EU is not and has never been a universalist project. Its purpose remains to be to unite
Europe, not the whole world.