Viewed through the lens of national self-interest, I’ve always thought of Turkey joining the EU as a general good, irrespective of what what it meant for Europeans. To some extent, I think fears about a changing European character or an immigration onslaught aren’t part of the US calculus for support, to say nothing of these fears being overblown. Having a moderate Muslim country as part of a Western transnational legal and trade framework is a boon to the world superpower. Of course, it’s the Europeans (or at least their elected representatives) and not the United States who will decide, so what’s best for America is somewhat besides the point.
With that in mind, I’m not sure how to think about a Turkey with a Muslim president (via Yellow is the Color). A president with strong religious credentials seems like a great partner for a Turkish EU induction. I often find European (and in this case, Turkish) secularization too aggressive, but much of that comes from living in a country where candidates openly profess their belief in God and no one tries to lead a coup. I accept that different concepts of secularization will be optimal for different countries.
Additionally, without narratives of long-term immigration and citizenship to fall back on, many Europeans are hostile to Muslim immigrants, and see their open displays of faith as an affront to European culture (witness France and their ban of the hijab). But Turkey bans the headscarf in government offices and schools too, and the Turkish military is hyper-vigilant on the secularization front. A Muslim president might upset powerful secularists within Turkey, causing internal turmoil that impedes integration, or spark a backlash against the country from right-wingers in Europe, making an invitation to join the EU even less likely. Turkey in the EU might be great for the US, but it seems increasingly harder to convince either Europeans or the Turkish that it’s also in their interests.