I had some success making filo pastry for apple strudel with a mixture consisting of flour, olive oil, lemon juice and white vinegar. The result was actually quite nice except the pastry was very delicate. I made the dough very thin using a pasta machine. I am wondering what would happen to the pastry if I used an egg. I am thinking it might be more crunchy... due to the egg introducing more protein fibers. What do you think?
In my personal experience, adding eggs to dough makes it softer, not crunchier. Adding an egg makes it an "enriched dough" that has the ability to hold onto moisture despite being baked.
Finally, while strudel dough is very similar to filo, it is not the same thing, which may be causing some confusion for you. Per one baking blog:
So the difference is simple, it's the fat! In the Puff Pastry the butter is folded in, in the Strudel dough the oil is part of the dough from the beginning and the Phyllo dough is brushed with oil before being baked.
So per your question, strudel dough sometimes includes an egg, whereas filo (phyllo) never does.
While it is possible to add an egg, it isn't a very promising idea. If what you like about phylo pastry is its traditional texture, then the easiest way to get it is to use the tried-and-known methods to make it, instead of doing random experiments.
So first about the egg: the yolk would be a no-go, it will make the pastry softer and less crunchy. Adding an egg white (or rather replacing some of the water with egg white) would be something to try, since egg whites do produce a drier result, but you will still be moving away from the standard phylo texture, even if you do achieve some kind of crunchy.
As for making a crunchy strudel, you don't need any complicated recipe. A standard phylo has flour and water, nothing else. I would certainly search for a recipe without oil if you have problems with crunchiness, since it softens the dough. The acid is also not needed - if you have problems with insufficient gluten development, just knead properly, rather than trying to adjust with acid. It probably won't hurt if you want to keep it, but my personal preference would be to choose a simpler recipe for learning the basics before switching to something with higher complexity.
Once you have made the dough, all you need for crunchiness is thin enough sheets (I would be suspicious of that pasta machine) and sufficient fat, both inside the dough sheet and on top of the rolled strudel (the taste is better if you place pieces of butter on top instead of sprinkling with oil). This ensures that the outer layers of the pastry get quasi-fried in the oven, making them crunchy. It is normal for the inner layers to stay soft, especially if you have a dairy-based filling or something on the wet side.