How can I make a soft Turkish bread?

So, I've checked some books (see below), and according to them you have two issues: (1) your finger dents aren't deep enough (puffing up), and (2) your oven is too hot.

All three of the recipes I found for pide bake it at 220C. Cooking it at 400C is likely what is causing it's more pizza-like texture. The oven in the video appears to be hotter than that, given that it's wood-fired, but we don't really have a temperature, and all three recipes I found baked it at 220C for between 13 and 25 minutes. Even if you don't drop all the way down to 220C, you might want to try lowering the heat in the oven, and taking the bread out sooner.

The pide recipes I consulted were:

1. "Pide" from The Complete Book of Turkish Cooking by Ghillie Basan;
2. "Pide" from Savory Baking from the Medditeranean by Anissa Helou;
3. "Tirnak Pidesi" from Istanbul & Beyond by Robyn Eckhardt.

Of the three, the last recipe is probably the closest to the one from the video. The first two recipes use yogurt and/or olive oil in the dough, which would tend to make it more tender. I don't recommend trying that in your Uuni, though, since olive oil doughs tend to catch fire at 400C.

Eckhardt's recipe, though, is just flour, water, salt, and yeast, plus an egg wash like the bakery. She even has instructions on a "stretched" version, like the video. She recommends "bread flour", which probably means high-protein flour. Her ratio is 4 1/4 cups flour to 1 7/8 cups water (American measurements), and she kneads it for 8-10 minutes, and rises it for 2 hours. She lets the dough rest for 15 minutes before denting and stretching, which if you haven't been doing it, might help with the texture.

Hope that helps; see if you can find a copy of the book.