When I have dough that fails a windowpane test I really feel it needs to go back in the machine, rather than be hand mixed. I have never, (and believe me I've tried) seen much improvement in a hand knead. In fact, oddly, it seems to make it worse. I've never quite determined why, but I've certainly tried a zillion times.
I don't break off a tiny piece to test. I test using a full dough ball size piece. Again this is from experience. Some try to make a little 1 inch by 1 inch square and if it looks translucent they think it's ok. To me test test is to see if the dough will go full size without breaking.
The truth is that after a few experiments you have to be able to advance to a different level of testing anyway. The dough may come off the hook not testing well, yet after 15 minutes it would test great. But by then you don't want to stretch a piece because once you stretch it out and roll it back up, it will never quite spring back the same for the pizza. So for now I'd make a bit extra and test that and toss it. But really, after a while you can just tell what it's supposed to be like at the hook and so you never test at all. The overall smoothness, satininess and blistering of the dough balls is the indicator that the dough is in great shape.
When the dough is done mixing, sprinkle a tiny bit of flour on it and feel it. It should be SOOOO soft and smooth, even if the areas that had no flour sprinkle were sticky and not so smooth.
This is a lot easier to show in person...