It's all about perspective and what is most useful to think about. You can assert there's an origin, or you can assert there isn't. Both are correct in different sense.
In some sense, you can view a "rational line" as a line whose points are identified by rational numbers. There's no need to have a specific starting point; I can call point A my origin, and you can call point B your origin. Because the rationals are a vector space, we can freely shift from your perspective to my perspective. If A-B winds up being 3/5, that's how much we have to shift our respective viewpoints by to look at the same point from the other persons' origin.
0 is a special number because of what it does for us in both addition and multiplication. It's also a fairly known concept; I start with nothing, and then I get one of something. It's natural, to us, to start from a neutral position.
In another sense, you could say there's a starting point. The Peano axioms are used to construct the natural numbers (0,1,2,...) from nothing; a set with nothing in it, and zero is the size of that set.
In that sense, 0 is the origin because the natural numbers are constructed from it, from which we get the integers. And then, the rational numbers are the X-Y plane whose elements are integers, whose form we give "x/y".