You raise some important questions. My experience with college students and this type of trip (secular version) is that a good handful have been motivated to do ongoing fundraising and donations once they’ve visited a community, or they switch to a service major or graduate degree, or they join the Peace Corps. Part of my challenge to them is, how will you pay back the expense and the carbon footprint of you going somewhere else? Another part of my challenge is, now that you know what to look for, where do you see it close to home and what are you going to do about it?
You’re making the assumption that all Christianity is the same. Your forgetting that a lot of these people are the types that go out to convert Catholics to Christianity.
They count as Christians when we are asked to help our Christian brothers and sisters, but count as the unconverted when doing proselytizing* work.
Not that proselytizing is always the primary goal of these missions. And, when even it is, it’s often working alongside the other Christians. The foreigners add an attraction for people who would otherwise not attend something Christian. I know that was the case in Mexico. The fellow Christians were not the target audience.
I know I’ve never seen a conversion story from a Turkish missionary be about someone who called themselves a Christian. It’s always about the Muslim who “sees the light.” The stories about the Christians is always about how something horrible happened to them because of their faith, or how they persevered and beat the something horrible.
*First time I’ve spelled that right the first time.
My niece went on one of these church trips. It was, as I recall, a two week trip. They helped build a clinic in Jamaica and as a side project they build a ramp for the house of a guy who was confined to a wheelchair. As far as I know, there was no proselytizing - they just did construction labor. Except there was one day at the end, after the project was finished, which they spent at the beach doing typical tourist stuff.
Now I can see the OP’s point. They could have taken the money they spent on flying to Jamaica and sent it to Jamaica to hire some locals to do the construction. But overall, the group did get the project built so a net positive good was accomplished.
One other thing she told me. As they were building the clinic, a lot of locals stopped by to watch them. So somebody asked their contact person why people were watching them. They were told “It’s because you’re white.” And they said “With all the tourists around, everybody must see white people all the time.” The contact said, “Sure. But nobody’s ever seen white people working before.”
True, they’re mostly the “bad” Christians – the Orthodox, or the Catholics. 
(Sadly, but in true Internet tradition, you typed “your” instead of “you’re” above.
Um, Turkey doesn’t have a large Christian population. Essentially all the Christians were ethnically cleansed out of Turkey during WWI and the immediate aftermath in the 1920s. (In fairness, the removal of the Christians in the '20s was part of a mutual exchange, Greece did the same with its Muslims).
Well, not large, but I’m guessing some of these people probably see Christians as a persecuted minority, or that Turkey is somehow like the Middle East, when it’s really not.