Beyond the ontological status of antinatalism, there is an important flip side to consider in the question of the practical implications of antinatalism i.e. what can be done to bring about or to engineer antinatalist outcomes, and having established these, the question of whether those methods will succeed. Indeed, it's one thing to ruminate upon these matters and yet another entirely to act upon them.
We consider first the case of what will happen if no action is undertaken, if antinatalism is espoused no further leaving the human race to carry on unabated. In the short term, people will continue to execute the same genetic program their millions of predecessors did; procreation and life alike continuing indefinitely. However, as nothing lasts forever we seek potential disruptions to this trend.
As for dangers within, it seems implausible that humanity could thinkingly or unthinkingly destroy itself. While the same myopic and selfish human nature responsible for procreation is the same that often ironically works in favour of antinatalism through over consumption and the degradation and destruction of the environment, humanity as it stands is far too resilient to be completely defeated by this mindless virus-like behavior.
If humanity does survive into the foreseeable future, disruptions to the status quo will likely come either from genetic engineering or from technology. If neither ultimately culminates in the destruction of humanity then they may have certain favourable outcomes. With genetic engineering comes the potential for the creation of organisms resistant or immune to suffering thus largely mollifying the adverse impact of procreation. Similar things may come from the technological front through abilities to manipulate conscious states and indirectly via the elimination of resource scarcity. In this case there is a possibility that antinatalism will be rendered largely obsolete disregarding possible breakdowns in these mechanisms.
Beyond these short term affairs, the current laws of physics coupled with cosmological observations suggest that the universe is bound for heat death; a relatively homogeneous state in which no usable energy exists precluding further ordered physical interactions. Because life as we know it depends upon the existence of free energy, such a universe would be incapable of supporting our lifeforms.
This outcome is favourable to antinatalists and more broadly to negative utilitarians yet the possibility of other universes existing cannot be discounted. In particular, as we are currently unaware of the mechanism which gave rise to this universe, we cannot guarantee that no other universes could arise in future, possibly rendering futile the efforts of antinatalists.
There are other speculative ends acting on the local scale that may be worth mentioning here; the possible instability of matter, vacuum metastability, cosmological topological defects, a misguided technological singularity. These have less global impact but could at least for a short time quash life on Earth.
Next we consider the remaining case in which antinatalists, either through collusion or though a collection of individual efforts attempt to bring an end to procreation either directly or indirectly. Is such an outcome possible? The evidence would suggest that the best possible outcome of the spread of propaganda is highly unlikely. As a case in point, we draw our attention to a similar moral position; that of vegetarianism/veganism. The parallels to antinatalism lie in both its support through basic human moral intuition and though its relatively widespread rejection. A key difference however lies in the fact that even though the tenets of vegetarianism are widely rejected, they are also widely known which leads us to conclude that even with greater exposure, antinatalism would suffer the same fate as vegetarianism. It seems that it scarcely matters how well reasoned a position is, humans are generally limited by their inability to perform actions contrary to their own interests.
Given this, the remaining issue is whether humanity can forcefully be brought to an end through the efforts of a small group. If they were to succeed, it would probably be through omnicide; mass forced sterilisation being far too problematic. But as it stands now, I'm not aware of any way in which omnicide is practical. Such plans would probably involve great ingenuity beyond that of which I'm capable in this brief essay.
Even if it isn't possible to completely end humanity, work could still be performed to partially mitigate the current harms of procreation. In accordance with the idea that the harmfulness of procreation is related to harms that life faces, antinatalist efforts would be better focused on the developing world. Programs like this could be suited for poorer people whose fertility could be purchased more cheaply and whose countries have less stringent law enforcement that might impede such philanthropy. Another possibility is to disregard humans and focus on animals instead. I suspect there is room here for more devious schemes to be conjured up in future.
I suppose I should conclude on a lighter note. A simple observation yields the insight that seven billion people can live life in their perpetual hedonic daze yet all it can take is one sufficiently motivated and capable individual to spoil the party for everyone. A converse is also true; that seven billion humans can be eliminated and yet a single surviving couple can ruin things for future generations. But let's not become too pessimistic, shall we?