Yeast dough won't go "stretchy"
Ok, this is going to be long. And you just wanted to fire up your oven and slap the sauce on the dough...but bear with me.


Gluten
The Holy Grail of elastic dough that can trap all these nice bubbles: CO2 from the yeast and steam from evaporating water. Fact is, gluten is basically a protein (ok, scientifically speaking not exactly, but close enough). If you use a low-protein flour, there simply isn't much gluten that could develop. So American cake flour isn't a good choice, but German 405 not necessarily either. Unlike the mineral content, protein content is not fixed for 405, it largely depends on the used wheat. Some mills have a higher content, some a lower. Apart from explicitly asking the manufacturer or simply trying out brands, there is no surefire way to tell. The reason the Italian tipo 00 contains more protein (and it does) is that most Italian wheats are high-protein breeds. So yes, using Italian flour can help. But so can changing the flour brand, switching to a slightly higher type (550 or adding a few spoons of 1050) or simply adding some gluten. The latter is available over the Internet or at your trusted local mill. No need to go "super-high" though. Patience and good technique are essential.
Humidity
Gluten needs water to develop. (Apart from the kneading etc., but more on that later.) A higher hydration dough1  means more water is available and the gluten strands form more easily. The math: with 500g flour and 250g water, you have a hydration of only 50%. Another 50g bump it up to 60%, 100g to 80%. As a ballpark figure, that's a good range for 405 or 550 flour. The second secret of the Italian tipo 00 comes into play here: it's milled somewhat finer than 405; due to the smaller particle size the dough can absorb even more water, following the old adage of "wetter is better" (not set in stone, but still...). You might want to check out the "stretch and fold" technique, which makes softer doughs easier to handle.
Time
Gluten develops over time. So a good, long kneading helps. But there is a another factor: resting (or proofing) time. Even with very little kneading gluten will develop over time, given sufficient humidity. You can do some light kneading, then let the dough rest for a long time at a rather low temperature with the occasional "fold".


So what can you do? Looking at your dough, yes, it is a tad dry and flaky and it tears a bit. But while you perhaps won't get an award-winning pizza crust, you'll probably be ok this time. But even while your recipe is pretty mainstream, a few tweaks might be advisable: 


Ensure sufficient protein content. (See 1.)
Aim for a hydration of 65% or higher (way higher, if you dare), especially if using tipo 00. (See 2.)
Significantly lower your yeast content. This reduces the "yeasty" taste your recipe should have. Yes, this means that your dough won't mature in half an hour, but the dough for the pizza you had today at your favourite pizza place probably was made yesterday, too. Simply let it rest and rise in your fridge. A handy side effect is that the enzymes in your flour get activated and the gluten develops, too. Good for texture and flavour. Bad for an impromptu pizza party, though. (See 3.)




1 Water-to-flour-ratio. See baker's percentage. In Germany "Teigausbeute"/"TA" (all-ingredients-to-flour-ratio) is more common.