Personally I like the hypothesis that states eventually all comms will be encrypted and therefore indistinguishable from white noise.
But we couldn't. In general, SETI couldn't even detect such a civilization even if it was in the star system closest to earth:
> If an extraterrestrial civilization has a SETI project similar to our own, could they detect signals from Earth?
> In general, no. Most earthly transmissions are too weak to be found by equipment similar to ours at the distance of even the nearest star. But there are some important exceptions. High-powered radars and the Arecibo broadcast of 1974 (which lasted for only three minutes) could be detected at distances of tens to hundreds of light-years with a setup similar to our best SETI experiments.
The OP is right, in order for their to be a paradox we have to assume that civilizations much more advanced than ours are using a particular type of tech (the kind we're looking for, like radio broadcasts), that their using much stronger signals that we have ever used, and that the time period they used these hypothetical super powerful radio signals happens to be at the right time for us to encounter them. IE, let's say they used them for 2,000 years before moving on to other tech, but the signals only reached earth between 1,548,000-1,546,000 BC, or 450,000-458,000 BC, etc.
People are making a set of very, very specific assumptions based on little evidence. Then they see that the evidence that is available suggests that it's unlikely that every single one of their assumptions is correct, and they respond by calling it a paradox.
Your quote relies on us detecting re-runs of "I Love Lzcwwdrrr", but our telescopes can reliably do spectrographic analysis of faraway stars and planets these days. We'd likely be able to detect things like unnaturally high levels of hydrocarbons in otherwise oxygen-full atmospheres, and other industrial by-products.
If Alpha Centauri had a planet with an industrial civilization, we might be able to spot city lights.
We don't know what "unnatural" levels of CO2 on other planets are.
> If Alpha Centauri had a planet with an industrial civilization, we might be able to spot city lights.
We do not even remotely have that capability.
See: https://en.wikipedia.org/wiki/Methods_of_detecting_exoplanet...
Milky Way radius is 100 000 light years.
If our galaxy had lots of civilizations that just learned radio in last 100 years - odds are quite high that we wouldn't be able to detect them YET, even if we had the technology to do so.
The OP made a rather extreme claim that 'the universe could full of earth-level societies and we wouldn't know'.
If all 512 G-type stars in 100 light years had earth-level societies, we would be able to detect their higher powered radio transmissions. Right?
Any ordinary radio transmission dissipates the square of the distance involved. At a certain level, transmissions become so faint they are indistinguishable from quantum noise.
You can find the math done on Quora (apologies for semi-walled-garden link). Remember, there's difference between getting a signal whose properties one knows and getting arbitrary signal - if there a strong GPS transmitter or something we knew, we could wait long enough and get micro-data. But arbitrary signal, no, information theory says no. (read all the answers together).
https://www.quora.com/How-far-can-radio-transmissions-from-E...
Do you dispute this claim from SETI? Can you provide a citation?
But, still, they sift the skies, at least we are trying right ? :D