If cultural evolution's effects are sufficiently strong, phenotypic effects due to nuclear DNA might be swamped by the effects of cultural variation. Perhaps egalitarian memes might act to eliminate variation at the level of DNA. Or maybe cultural tags and tribal identifiers evolve quickly and generate so much phenotypic variation that the effects of nuclear DNA are swamped. In either case, population bottlenecks at the level of DNA might not matter much.
On balance, it seems unlikely that cultural variation makes the effect of recent population bottleneck on the variation in human DNA irrelevant. A good rule of thumb is that around half the variation in a typical trait is down to DNA. Cultural variation is important, but not so important that DNA doesn't matter.