“And the church most definitely takes credit and includes these amounts in their annual charitable contribution report.” Prove it. They don’t. Every article, including those from the Church point out that the Church facilitates the donations? They don’t take credit for them. Have you ever read their 990?
The Church covers the administrative costs of RAISING the money. That is where the VAST majority of costs arise. Sure, the organization still has overhead costs, but every dollar of raising the money (by far the most expensive part of the administration) is covered by the Church (including merchant fees), which allows 100% of your donation to go to the cause. (It doesn’t mean that the Charity for whom the money is raised won’t squander it, but it makes your dollar twice as effective because there are no costs in raising it.)
https://www.washingtonpost.com/religion/2018/12/24/buy-goat-or-soccer-ball-mormon-vending-machine-charity/