Fat can inhibit gluten development in things like cakes, where the fat is allowed to coat the flour before other liquids are added. In things like bread dough, it slows down the gluten development a bit, but since the flour has already come in contact with water, it doesn't stop it. In the context of your phyllo or strudel, the oil serves a couple of purposes. First, oil in a dough helps to keep it from drying out which is very important when working with such thin doughs. Second, adding oil to a dough can help to make it easier to handle (less sticky).
The resting period of your dough both allows enzymatic reactions in the dough to further develop the gluten, and it allows the gluten to relax. You can think of the gluten as strings in your dough. As you knead the dough, you're winding these strings tighter and tigher, but when you let it sit, they can begin to stretch and loosen. This relaxation makes the dough much less "tight", so that it can be rolled out more easily without springing back as much.
While many bread and pastry products do depend critically on the formation and management of gluten from wheat flours, this is not universally true.
Some types of pastry have structure dependent more on the starch networks which is the other major component of wheat flours; the texture and properties of these pastries is often dependent on the gross mechanical manipulation of the structures, depending on how the product is manipulated.
American style pie crust is a case in point. Flaky pie crust is generally manipulated in a manner which minimizes gluten production: low hydration, resting periods, and minimal mechanical manipulation.
The structure and flakiness emerges because the dough is essentially a series of butter flakes or bits, coated in starch. When baked, the starches gelate and steam from the dough pushes the butter pockets apart before the butter is fully melted and integrates into the dough structure. This is not dependent on gluten formation.
Taking La Cucina Italiana's pasta frola recipe as typical, it appears that pasta frola has more in common with flaky pie crust than it does with bread or puff pastry, where gluten formation is key.
The egg in the dough will contribute mostly water (eggs are 75% water). The remainder is primarily proteins (from the albumen in the egg white, and fats from the yolk).
This will somewhat tenderize the final pastry, and contribute an eggy flavor. It certainly does interfere with gluten formation, helping prevent a tough outcome, except in this application, gluten formation simply is not key. It will also contribute to the final color, both from the yellow pigments in the yolk, and by slightly promoting browning.
Best Answer
Flour (+ water, either directly or from other ingredients such as egg white) gives the pastry structure. As you knead the flour, the gluten network develops and results in elasticity. When cooked, water evaporates from the dough leaving a rigid gluten skeleton.
Fat does not mix with water and thus stay in blobs in between the gluten network. This weakens the gluten structure, making the pastry crumble.
For this reason, you often see in recipes for shortcrust to avoid overmixing the fat. Buttery biscuits such as shortbread crumble more than crackers, which have more water. The same rationale applies to puff pastry. The thin fat layers are impermeable, so water steam gets trapped and exercise pressure upwards, lifting the pastry up.
Quoting @GdD in the comments, the punchline is:
more fat = crumblier, less fat has more structure.