So, it's pretty clear that tone holes on a wooden flute take up quite a lot of space, but it may be hard to visualize exactly how there would be a lack of space for the C# trill -- so we thought a photo refresher comparing metal flutes with and without the C# trill might be helpful. In the photo below, we see the two flutes next to each other. There is a yellow arrow pointing to the key cup with the C# trill.
Now, looking at the photo below of the back of a wooden flute, it definitely becomes clearer. The C# trill tone hole and key cup would simply be too close to the thumb keys (because it would be in the same place it is on a metal flute).
So, there you have it! The reason why there is no option for a C# trill is because of space and location -- not enough room for the C# trill tone hole and key cup, and definitely not a good spot on the wood for this, either.