Who exactly paid for it?

Interesting, and a bit bizarre.  

However, I disagree with your formulation "Contributions to a Church to support a child's mission trip are not charitable contributions".  The issue in Davis was that the church established the amount but the money was paid directly to the missionaries.  I would say that if the church solicits donations for a mission trip and the parent write the check to the church, which now means that the church has control over the money (and could, for example, cancel the trip and use the money for something else, or drop the parent's child for behavior reasons) then it is still a deductible donation to the church.  Paying the child's expenses seems to be non-deductible (again, bizarre) but I would argue that donating money to the church would still be deductible.

(It's an arms' length transaction. Similarly, if you give money to a poor family at christmas time, it is not deductible, but if the church board announces "we have identified a needy family, please put your donations into the drop box in the back" then it is deductible because it is a donation to the church rather than a person––as long as the church board decides who the beneficiary will be a not you the donor)

I believe your comment about the Church designating a family at Christmas time to be very much on point.  However, to apply that to this situation, the Church would need to designate the child.  Donating to the Church with the understanding that it is to be used for a specific child would seem to be insufficient to claim a deduction.

Perhaps OP will come back and clarify what he/she meant when they said ". . . through the church."

Well, if the church has a general fundraiser for a youth group mission trip, should member A with no children get a deduction while member B who has a child in the youth group is denied?

As distinguished from "anyone who wants to go on the trip should write us a check for $575.32 to cover their expenses."

So long as neither Member A or Member B's contribution is contingent on any specific child going, I would give either the deduction.

Yes, with qualifications.  The main purpose of the trip has to be charitable service.  Also, any costs for recreation or personal activities are not deductible.  

The IRS says: Generally, you can claim a charitable contribution deduction for travel expenses nec­essarily incurred while you are away from home performing services for a charitable organization only if there is no significant element of per­sonal pleasure, recreation, or vacation in the travel. This applies whether you pay the expen­ses directly or indirectly. You are paying the ex­penses indirectly if you make a payment to the charitable organization and the organization pays for your travel expenses.