Common sense tells us that effect of gravity moves at the speed of light

This appears to be an open question; it wasn't until 2002 that they were fairly sure and very recently (2017) that they established that, "assuming a delay of zero to ten seconds, the difference between the speeds of gravitational and electromagnetic waves, vGW − vEM, is constrained to between −3×10−15 and +7×10−16 times the speed of light."

I would have thought it was easier to apply common sense.

The light we see from the sun arrives from where the sun was 8 minutes 20 seconds ago (assuming for simplicity the earth is stationary and the sun moves round it).

If you measure the direction in which the earth is being pulled by the sun, you'd establish that it is being pulled towards where the sun was 8 minutes 20 seconds ago.

The light meter and gravity meter will be pointing in exactly the same direction; if not, there'd be all sorts of apparent weird wobbles (which would make measuring distances and so on a lot easier, as it happens).

And well done to these chaps, while we're on the topic. Instead of chasing non-existent Dark Matter, they are actually measuring and observing exciting stuff that actually happens.