For the analogy to apply, imagine you have an infinitely large sheet of paper, with its edges at infinity.
If I asked you to draw the line segment with its first of two end points (one being the “origin” of Time, “Big Bang”, etc., and the other representing the present moment) at the centre of the page (the other end point being radially away from the centre) how would you find this centre?
In an infinite by infinite plane it simply doesn't matter because the [non-existent] edge is an infinite distance away from any given point. IOW, the origin is arbitrary and may legitimately be said to be anywhere -- thus arbitrarily deciding/defining that the "start of time" point is the origin is valid.