Briefly, until the Earth was gobbled up for material to make new self-replicating interstellar probes. Evolution would ensure that the probes would be voraciously reproducing as quickly as physically possible.
The numerical argument that the age of the galaxy gives sufficient time that it should be well-covered with probes seems reasonable.
However, extending that argument to "...and the various probes will have then competed in a Darwinian struggle leaving only a few voracious survivors causing a galactic-scale grey-goo scenario." does not seem especially well supported.
