First off, congrats on the "alpine route"! I think you'll find after doing more of these that the slope pictured will feel a lot more comfortable for you.
Ice axes are not just for ice!
You say
Since it is rock I'm not sure how useful it would be to have a pickaxe ready to arrest one's fall.
Actually, it could be quite useful, especially because there's also a lot of dirt and sand involved. The standard textbook Mountaineering: The Freedom of the Hills has this to say about scree:
Ice axes are helpful; the technique on scree is similar to that on snow. Nonetheless, be aware that scree can sometimes consist of only a thin, ball bearing–sized cover over large rocks. If there is vegetation on the slope, avoid setting off a scree slide that can damage the plants.
In many cases, you can self-belay by keeping a firm grip on the axe and driving the spike into the dirt uphill of you as you traverse. A fall can be arrested much like on snow, albeit likely more painfully. This answer has some further information that you might find useful.
I also really like trekking poles for scree, except for the "ball-bearing" situation described above. Using two poles negates the use of an axe, but one pole and one axe is also a possibility.
However, your safety in this situation primarily depends on your footwork.
It can be difficult at first when transitioning from simple trail hiking to mountaineering. On a trail, each step is usually solid. If you wanted to stop and stand in any particular place it would not be much of a challenge. In a more scramble-like situation such as this, you must maintain dynamic footwork, moving or being ready to move to the next step before the previous one has had a chance to settle. Note that this does not imply rushing or scampering, but rather executing a series of fluid movements to shift yourself between positions of balance. It takes practice to develop a feel for various types of terrain, so if this one felt uncomfortable then find a more mild route to practice on before returning.
And I always recommend wearing a climbing or mountaineering helmet in these conditions. If you do slip, a helmet could save your life.