https://en.wikipedia.org/wiki/Timeline_of_the_evolutionary_h...
We need to start developing meaningful working hypotheses and make judgement calls instead of waiting till we have definitive proof. This was similar to how basically no one was surprised that every star out there has planets around it yet we were writing sci-fi under that assumptions for basically a century before definitive proof.
The bigger question is: how did the first Protozoa form? The usual primordial soup theory of random chance?
One good thing about this theory is that if there are 10^7 habitable planets at any given moment, each of them are "trying to produce life." So the chance that life gets to Earth increases multiple fold.
One problem for this theory is that if it takes a thousand years for Earth to get protozoa from another planet, then we would expect to have gotten multiple different types of Protozoa. But there is basically no variety on Earth: it's all carbon, and 99% uses oxygen (a few use sulfur).
However, if we assume a seeding event happened, what's to say it couldn't happen on a lot of planets around the same area at the same time? The Fermi paradox is too speculative to say we aren't the only ones.
One thing that's missing is a seeding mechanism. Can an asteroid eject enough debris for some of it to reach escape velocity on a planet like ours? Did these protozoa come from a very different (but chemically compatible) environment?
So the Star Trek episode where they discover why all the species around the galaxy all have DNA based genetics is probably on the money if you ask me!
Ok, even if this mechanism of fertilizing planets was indeed functional as you suggest, we still don't see any evidence of intelligent life out there in our galaxy. Surely, out of billions of planets and billions of years time there would emerge at least a couple of civilizations capable of colonizing a significant part of our galaxy. There's certainly a slim possibility that we may be the first of such civilizations, but I'd be surprised to find out that humanity is THE smartest and most reasonable civilization out of billions others that our galaxy was capable to produce so far.
Another possibility is that the window in which life following similar intelligence trajectory as us will actually broadcast in measurable frequencies for a short while (centuries) before rapidly moving on to the next phase of their evolution (who knows what it is? Star child? Transcendence? Warp dimension) so we only have thin shells of radio waves expanding from each planet that developed intelligence making it impossible to detect.