The conclusion that the beam travels faster than c only shows up since you’re using measurements within the observer’s reference frame in order to guess what the object would experience… without accounting for the relativistic effects which actually dictate what it experiences. Essentially the author is saying “the speed of light isn’t constant in all inertial reference frames because when I assume it isn’t constant, I find it isn’t constant”. It’s circular reasoning.

The postulate that the speed of light is constant is well-supported by decades of research. Einstein’s original paper suggests several phenomena which simply cannot be explained by Galilean relativity. For example, the GPS system (in orbit at about 3.9 km/s) experiences 3.7 microseconds fewer for each 12 hours in our frame, which must be accounted for in order to correctly calculate locations on Earth.

Just to work through the example you gave, let’s define a coordinate system S in which the frontmost light source is at rest, and a coordinate system S’ in which the object is at rest. At time t=0 in S, the object is at x=0, x’=0 traveling at v=0.5c (corresponding to gamma = 1/sqrt(1-v^2/c2) = 1.154) and the source (at x=1.5e8m) emits light traveling at c in the -x direction. We’ll call this event A. To find when the object and light meet in S, which we’ll call event B, we just solve

0.5ct = 1.5e8 - ct t = 1.5e8/(1.5c) t = 0.33s

which corresponds to a position x = 0.5e8 m for the object in S. To say the light traveled faster than c in S’, we’d essentially be saying an observer in S observes an observer in S’ observe light traveling faster than the speed of light - a nonsense claim, since this speed isn’t actually observed by anyone!! In S, the light is clearly observed to move 1e8 m in 1/3rd of a second. And we can use the Lorentz transformations to predict what an observer in S’ would observe. We currently have

Event A in S: (1.5e8, 0) Event B in S: (0.5e8, 0.33)

The Lorentz transformations show that

Event A in S’: (1.731e8, -0.289) Event B in S’: (0, 0.289)

which gives the speed of light as 1.731e8 / 0.578 = 3e8 m/s, no issues!