Remember that Star Trek was able to predict the flip phone, tablet computer, Bluetooth headset, and other modern technology that was inconceivable to the everyman of the 1960s.
Star Trek also predicted that no-one would ever actually obey the prime directive. But in any case the prime directive is largely irrelevant. We don't need aliens to park on the White House lawn in order for us to see them. A true stealth/cloak system will never exist - using one would render you blind and incapable of propulsion. Any alien visitors must be visible, and any communications must be interceptable. Given how good we're getting at spotting small, inert rocks that come too close, how easy is it going to be for a spaceship intent on studying us to hide?
But in any case, the whole question is largely irrelevant. The important point is this:
Depending on what numbers you plug into the Drake Equation, estimates suggest that there may exist hundreds of thousands of civilizations in the Milky Way.
No estimates using the Drake Equation suggest anything. We simply do not know any of the numbers. If someone suggests there are hundreds of thousands of civilisations in the Milky Way, that is not an estimate, it's simply a wild-ass guess based on absolutely nothing. The Drake equation tells us what variables we would need to know in order to make such an estimate, but with our current level of knowledge it does not actually allow us to do so. There is therefore no paradox. If you guess that there should be lots of civilisations wandering around the place but we don't see any evidence of them, the conclusion is simply that your guess was wrong
. A paradox occurs when two contradictory things both appear to be true. It is not a paradox when a prediction simply turns out to be wrong.