Persuade me that anarchy is viable in today's world

First of all, there are many different types of anarchy. While anarchy itself in general is just the lack of state, there are many ways the stateless society can self-organise. The most popular types of anarchy, as far as I know, are anarcho-capitalism, anarcho-syndicalism, anarcho-communism and "absolute anarchy" - the latter being essentially a human version of animal kingdom with no rules and norms.

Absolute anarchy obviously would be easily defeated by any organised military machine. Collectivist forms of anarchy, such as anarcho-communism and anarcho-syndicalism, have powerful mechanisms allowing people to unite in the face of external threat - but the economical systems themselves are so inefficient, that they are likely to fall even after that.
For anarcho-capitalism, it depends on how wealthy the society is. A society wealthy enough, in which each household can afford a lot of weaponry and automated defenses, and which funds powerful modern private security companies, can funnel their resources into a temporarily formed military machine tasked with resisting the external aggression and nothing else - and in the end it functions just like a strong military, only built on a much more powerful decentralised economical base.
On the other hand, if the anarcho-capitalist society is only starting out and is still relatively poor, then it can be crushed by a strong foreign state military.

I see anarchy establishing itself according to the Swiss and Singaporean economical-diplomatic models: "Make yourself so useful to everyone else that conquering you will make the conqueror worse off". An anarchist society can feature extremely investment-friendly economy, and once everyone has invested enough in it, conquering the society becomes unprofitable to them, as their investments stop giving returns. The anarchist society, in the meantime, can use the investments to boost its economical growth. At some point, the individuals become strong and prosperous enough to be able to withstand most invasions.

Bear also in mind that the historical examples you are citing are from the past. Up until the recent few decades, conquest was seen as a legitimate political action. It is no longer the case today: attacking another state with the goal being its conquest is no longer considered acceptable, and anyone who does it without having a very strong casus belli will face international ostracision sufficient to deter everyone else from seriously considering doing the same. Modern wars are typically short and non-expansionist, having some specific goals in mind, such as demotion of the dictator or defeating a terrorist organisation. The last time anyone conquered anything was Russia 2014 taking over Crimea, and the economical sanctions it had to face halted any further expansion indefinitely.
I do not think we have to fear being conquered out of the blue today, no matter who we are, as long as we are an integral part of the international community - and as such, the anarchist model that was not very viable in the past may be viable today.

Finally, the emergence of the state inside the anarchist society can be of concern. To that, I can only say that it takes a very developed and enlightened society to prevent that from happening. The vast majority of individuals in the society must exhibit strong anarchist views, and be willing to fight any entity that attempts to destroy the anarchy. If enough people deeply oppose the existence of any state, then I do not see how the state can appear without strong support of the population.