One author who explicitly avoids assuming the Hausdorff property is Serge Lang, "Fundamentals of differential geometry", Springer 1999, 2001. In Chapters II and II, he does not assume Hausdorff. He introduces the condition at the beginning of Chapter IV. He says:
We see no reason to assume that $X$ is Hausdorff. If we wanted $X$ to be Hausdorff, we would have to place a separation condition on the covering. This plays no role in the formal development in Chapters II and III.
It's fairly clear from the literature that the reason for assuming Hausdorff is because it is always true for embedded and immersed manifolds, and that's what differential geometry was about in its first few decades. If you don't have this condition, you make possible not just the "line with two origins". You can also have two closed unit intervals $[0,1]$, for example, and a whole Hilbert spaces of unit intervals if you like, and you can have any number of "foliations" from individual lines going off to create vast complicated networks of lines. And then in $n$-dimensional spaces, you can get an astonishing variety of spaces with topologically closed regions containing "bubbles", and in space-time, you can get bubbles opening and closing in astonishingly complicated ways over time.
You can regard non-Hausdorff manifolds as either a huge opportunity to construct physical models which have mixed states in the quantum sense, or you can regard it as a nightmare to be avoided if all you want to do is geometry of manifolds embedded/immersed in Euclidean spaces.
Just one more little point about this issue.
It's fairly obvious that you can't metrize a non-Hausdorff manifold. What's not quite so obvious is that it's difficult to even pseudometrize it. I was thinking that you can put a pseudometric on the "real line with two origins" by letting the distance between the two origins equal zero. However, there is a very easy theorem which says that any pseudometric (i.e. a metric which does not require equality of two points if their distance equals zero) must be a metric if the topology is $T_0$. But every locally Cartesian space is $T_1$, which implies $T_0$. Therefore every non-Hausdorff locally Cartesian space is non-metrizable and also non-pseudometrizable. (I'm assuming that the set contains more than one point!)
This makes these spaces somewhat useless for Riemannian geometry. You can put differentiable structures on them, and maybe you can put affine connections on them, but the familiar Riemannian metric can't work. (Any affine connection would not be a metric connection for any metric.) So there's another reason to reject non-Hausdorff manifolds.