If there was any time the human population was very small, fixation was likely inevitable as discussed in Neutral Evolution for Newbies, Part 2. The time for required for fixation in a population according to standard population genetics is approximately 4 Ne, where Ne is the effective population size.
For example, for an Ne of six individuals, the approximate time to fixation using this approximation is:
4 x 6 = 24 generations
In 24 generations, assuming they don’t die from inbreeding depression, everyone will be pretty much genetically identical according to standard theory. Even if their differences were huge (maybe millions of nucleotides), they should fix in 24 generations using this approximation.
In contrast consider the current human population of 7 Billion, and suppose the effective reproductively viable population is 1.5 billion. Using the approximation, the time to fixation with Ne = 1.5 billion is
4 x 1,500,000,000 = 6 billion generations
Assuming a reproductive generation is about 20 years, that’s about 120 billion years on average to fix a new trait!
Now given gametic mutation rates are estimated in the ball park of 3.0 x 10^-8 per nucleotide position per generation, let’s do some math as to how much an individual nucleotide position might get overwritten with mutations in 6 billion generations:
3.0 x 10^-8 (mutations per nucleotide position per generation) x 6,000,000,000 generations = 180 mutations per nucleotide position
Which looks non-sensical because it essentially says in the time it would have taken to fix a single point mutation in a given nucleotide position, it will have been scrambled 180 times over anyway, so again as I said before, we can’t blindly follow the formula that says fixation rate equals gametic mutation rate. I gave other conditions under which that claim cannot hold in Fixation rate, what about breaking rate?.
Now if we are dealing with selection, large populations actually help fixation of traits even under weak selection, but how long will that take, and will it even happen for geographically spread out populations that don’t migrate much?
In the case of anti-biotic resistance where a billion bacteria could be killed off for every resistant strain in the first phase of evolving a resistance, natural selection works quite well in fixing a trait. But such forms or truncation selection will not happen in the current human population barring some sort of epidemic that kills of all but a few of the 7 billion living individuals on the planet, at which point we might suppose the human race might as well be extinct should that happen.
If the fixation time under relatively constant population and weak selection takes too many generations, then for similar reasons like the neutral case, the genome would essentially be scrambled any way, so what little is gained by fixation by selection is lost by mutation accumulation. This scrambling is illustrated with the Poisson distribution in Fixation rate, what about breaking rate?.