What kind of homemade sandwich bread or wrap lasts longest?
Firstly - shop-bought "packet" bread will keep for several days because it contains a lot of preservatives.

In his excellent book Bread Matters, Andrew Whitley claims that home-made sourdough breads with very long rises have better keeping properties than home-made bread made with baker's yeast and short rises. This, he claims, is because the sourdough yeast cultivates a culture of friendly bacteria and an acid environment, all of which become natural preservatives. My experience seems to confirm this.

Bread recipes containing egg, oil or milk tend to keep for longer than those without.

As @SAJ14SAJ says, bread freezes very well. Loss of flavour in the freezer is not a commonly recognised phenomenon.

One option is to slice a loaf of bread, bag it and freeze it. Take as many slices as you need at a time. It will defrost very quickly at room temperature, due to its low mass and high surface area.

Another option is to make rolls, part-bake, and freeze. When you need them, take as many rolls as you need from the freezer, and place in the oven, to both defrost and complete the baking. You will get fresh baking smells and flavours from this.

I suspect that home-made unleavened breads (pita, chapati, tortilla), without added preservatives, will not last long out of the freezer. Their advantage is that with no rise, they are so quick to make that they can be made fresh on-demand.