Travel Times for Seismic Wave Types - Why reflected t-x plot curves
This is conceptually relatively simple if you look at where the reflection (and refraction) happen, and in fact you have articulated the key concept in your question:


  The reflected wave ...Over time it becomes more horizontal and therefore more like a direct wave which takes a shorter and quicker path across the surface. 


This is absolutely correct. So if you are close to the seismic source (distance a, where a is small), the direct wave is quickest, as the reflected wave has to first travel down to the boundary before being reflected back. It travels a little over 2b, where b is the depth of the boundary layer. That "little over" depends on the angle, which is related to the distance a.

As an example, look at the image below, from this presentation on reflected, refracted and diffracted waves. The 2nd of the reflected waves is not much larger than the first, however the direct wave has travelled 4 times as far.



If a is much larger than b, then that 2b portion becomes less and less significant, and in fact you can say the distance travelled by the reflected wave tends towards a.

So the graph does not show that the velocity of the wave is slowing down, more that the distance relationship with the angle of reflection tends to an asymptote.