Now that is a good argument that doesn't miss the point. My priors would say it's not even "a few centuries" - I'd expect less than one earth-century on average, with most of the variance due to the particular economic variations and social phenomena derived from the details of the species.
Without any other information, it is reasonable to place the average to whatever time it takes us (probably a bit over a century), but I wouldn't put a lot of confidence in that figure, having been obtained from a single data point. Radio visibility could conceivably range from a mere decade (consider that computers could have been developed before radio -- had Babbage been more successful -- and expedite technological advances) to perhaps millennia (consider dim-witted beings that live for centuries and do everything we do ten times slower).
Several differ...
"The Planet-of-the-Apes Hypothesis" Revisited --Will Intelligence be a Constant in the Universe?