In the Hulse-Taylor binary pulsar system, the orbit of the two neutron stars results in the warping of space time causing the pulses to arrive earlier and later because of the longer distance travelled when the pulse crosses the gravitational field. How does this prove that gravity moves at the speed of light?
You'll have to forgive me if I misinterpret anything, I just learned about this binary system now and have only done some preliminary research. However, from what I can tell, the delay in signal from gravitational warping is not the evidence for the speed of gravity that this system provides.
Instead, this binary pulsar system demonstrates a distinct orbital decay. That is, the periapsis (or to be pedantic, the periastron) distance decreases noticeably over time. This is predicted by general relativity, which states that two bodies like this should lose energy in the form of gravitational radiation (all information your link provided). In fact, the amount of orbital decay predicted and that observed matches to within half a percent (part of the reason why they got the Nobel prize for this).
This makes it extremely likely that gravitational waves are a thing. Of course, you couldn't have gravity waves if the speed of gravity were infinite. So, at first glance, it certainly seems to confirm that the speed of gravity is finite. Furthermore, the amount of energy radiated away with gravity waves is dependant on the speed of the wave. The theories relate the radiated power as proportional to $1\over v^5$, where $v$ is the speed of the wave. It turns out that if you make $v=c$, you get the exact amount of energy radiated away as what corresponds to the orbital decay of the system to within no more than $1\%$. So it would stand to reason that the speed of gravity is likely the speed of light.
No messing around with the delay of pulses due to spacetime warping is necessary.