Pizza dough is too soft to stretch well
It actually sounds like your dough is good, your description is just what I'd look for in a dough. It may be your technique which is the trouble. It's takes practice to hand stretch dough right, you could try using a rolling pin instead. 

You should be able to stretch your dough very thin and still have it hold together. If your dough is breaking easily then you may not have enough gluten development. If you still want stiffer dough then you could go for a higher gluten content, or go for a lower hydration level, say 70%.

EDIT: One thing I've found with pizza dough is that if you try to stretch it too far too quickly it will be uneven and prone to breaking. I've found that if you stretch it part of the way then leave it a few minutes it will relax a bit and be easier to work with.