Earth has been screaming it has a very likely biosphere for at least 500 million to a billion years. To anyone with huge space based telescopes.
So why no visitors? If there had been, we wouldn’t know. Any probes that dropped into our planet any further back than a few tens of thousands of years (and less if they landed in a hot wet region) might be gone by now. They’d have been eaten by corrosion and mechanical erosion and eventually by plate tectonics.
They also likely would have been small, meaning even if they got fossilized we’d have to get super lucky to find one. The energy required to accelerate something to meaningful fractions of light speed and then decelerate at the other side means a probe is probably an orbiter the size of a basketball and then a little drone the size of a golf ball or something.
We might have had dozens or hundreds of little visitors over the last billion years and we’d never know unless we got real lucky.
Flyby missions are also likely due to the physics. The energy for slowing down might instead be spent just going faster to get results faster. The probe just streaks past at 7% the speed of light and takes a bunch of pictures and measurements.
I consider this a disproof of the "dark forest" hypothesis. That doesn't affect the Three Body Problem as a work of science fiction at all; science fiction gets to make a few assumptions and then ride from there. But if it were advantageous to pre-emptively nuke anything that looks like a threat Earth should have been toasted hundreds of millions of years ago.
(And no, the dinosaur asteroid was not it. If an alien species is going to destroy Earth, they will destroy it, not slightly inconvenience the biosphere.)
It did take using the sun as an amplification source.
Well, if there was a "Three Body Problem" civilization looking for a new home, we have not been inconspicuous.
I think we can rule that out.