Yes, you are correct that you do not need to enter the 1099-Q ESA distribution in your taxes if it was all used for educational purposes. Here is a TurboTax article about 1099-Q distributions.
The ESA distribution was tax-free since it was used to cover all of your education expenses. You stated that the expense was exactly the amount of the education expenses. There is no amount left over to enter on the tuition expense screen.
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