I very regularly make tortillas myself using a very similar method, and had exactly the same problem. As I'm trying to cut down on the amount of fat used, I have been putting less and less in recently, but I'm still getting them lovely and soft. Here is my recipe:
60g of flour per tortilla (plain white works well, but I've had good results with brown flour too)
Water (In my experience, the temperature is irrelevant)
- Add water in small quantities and mix until the dough comes together, but is still quite dry (if it's too wet then you can knead it until all of the water is taken in, but try to keep it on the dry side)
- Tear up the dough into little chunks and splash on a glug of oil (I'm afraid I can't be more specific about this as I never bother measuring it, but I don't put in much)
- Combine the dough again, and knead it for a minute or two. This normally produces a smooth, pliable dough
- Separate the large ball of dough into portions
- On a floured surface, roll out each portion to the size of the frying pan (normally about 1-2mm thick)
- Dry fry on each side for a few minutes, until bubbles are appearing and none of the dough is visibly raw
- THEN (and this is the important step, as when I used to make them until this point, they also used to solidify) once fried, put the tortilla into the oven at a low temperature (50 degrees C) between two sheets of silver foil. What this does is keep them warm, and more importantly, keep the moisture in the tortillas. If you just put them in the oven (or leave them out) without the silver foil, all of the moisture will evaporate, and it'll become brittle, and not very wrappable.
I've have a very different experience to the other answer, where varying the amount of oil doesn't really change the texture that much, but can change the flavour (and their relative healthiness).