How is that different from the Aes Sedai, who do just as bad things to nations without even holding the level of responsibility that the Children uphold in Amadicia, and make the lives of men with innocent talents hell?
Or are you so knee-jerk supportive of Aes Sedai that the idea of a meritocracy dedicated to fighting the Dark One holding superior authority to an inbred aristocrat offends you? And that stuff about them torturing Darkfriends on mere suspicion is empty rumor repeated by people hostile to the Children. When do you ever see that attitude except in a few aberrant individuals like Asunawa & Valda who are shown to be out of step with the mainstream Children? The readers' hostility can generally be credited to the PoV trap, since our first view of the Children came from the perspective of a man held prisoner by them for two murders of which he is unquesitonably guilty! There is no getting around the fact that as bad as Egwene is, technically, Perrin is the worst of the Two Rivers folk, as he is the only one to have committed actual, unjustifable murder. The hostility he experiences at their hands is both deserved and rather mild as far as that goes. Not only that, they were playing his case by the rules. No one in the world would have found fault with them if they had killed Perrin on the spot (as he himself does later in the books on far less evidence), but they were still taking him along for a formal trial and execution, despite the hardship it was imposing on their rate of travel.

Beyond their justified hostility to Perrin (who later helps an Aiel murder a bunch more of them for no good reason, in order to facilitate and cover up his unilateral and unjustified decision to free Gaul), the only cases we have seen of the Children actually hanging anyone or committing any atrocities were Darkfriends infiltrating them (and thereby proving them right to be so suspicious - or do you hold the White Tower to blame for the Black Ajah's actions? ), or else actually hanging Darkfriends. As for their making wisdoms' lives miserable, don't the Kin prove how easy it is for channelers to masquerade as such? For the most part, it is revealed that all the Children do is warn people about the dangers of Darkfriends, whereupon the iniquitous villagers use that as an excuse to get rid of people they dislike.

As for their seemingly abrupt justice, why are they the only people in WoT (except for maybe the Seanchan) held to strict real world contemporary standards? We see that in other nations, the word of a high-ranking noble is sufficient to convict a person of capital crimes. Elayne arrests Mili Skane/Shiane based solely on her association with people she personally knows to be Darkfriends, and arrests Daved Hanlon/Doilan Mellar on similar grounds, despite her commitment to justice and proof of innocence. When have you EVER seen Whitecloaks in the series arrest people on as flimsy (to modern standards) grounds as "a known criminal claims to have seen him go into a house where a woman was addressed by the first name of a known Darkfriend and described in one sentence to resemble another Df?" Or in Cairhien, when Colavaere was arraigned purely on Dobraine's oath, and convicted & sentenced at Rand's will? How many times in the past year have you yourself raved with praise for Egwene for trying women based solely on the written confessions of a proven liar & self-excusing Darkfriend? The point of all this is not to condemn those characters, but to show that surprisingly thin evidence by our standards is accepted as proof for capital crimes, and the sentences in even moderate countries like Andor are rather harsh for even lesser offenses. A pickpocket who is not even remotely accused of harming so much as a fly can be mutilated and eventually executed for repeat offenses, in Andor.

By the standards of THEIR contemporaries, the Children are not nearly as harsh or hasty as you make them out, and the one objective view we have of them doing their "witch-hunt" stuff turns out to be a well-deserved execution of a Darkfriend cell, and inadvertant rescue of Morgase from falling into their clutches.

The question about the Children of the Light is not "How can they exist?" but rather, in a world rife with Darkfriends and tangible threats from the Shadow, which have on at least two occasions plunged the world into barbarism, should be "Why isn't there anyone else like them?" A world that treated terrorism as cavalierly as the world of WoT treats Darkfriends would not have the SAS, DSG-9, the Mossad, Green Berets or Delta Force. The FBI & other investigative bodies would not have special counter-terrorism divisions or bureaus. There would be one extranational counter-terrorist force in the world, and that a privately funded army which constantly had to deal with harrassment from the UN and various national governments and reflexive criticism and hostility in all media. Members of this force could be murdered for the crime of defending themselves against a rabid dog that attacked them at when stopping at a gas station, and everyone would side with the murderers! Meanwhile, terrorists would have thoroughly infiltrated every government in the world, frequently murdered the Secretary-General of the UN and other heads of state, and wiped out Western Civilization twice in the space of 1200 years, stalk people of interest to them across the length and breadth of a nation, infiltrate military bases and break out their accidentally captured comrades and steal crucial technology almost at will, with no more organized efforts to stamp them out than the occasional lynch mob!

The only reason for the absurd lack of efforts to quell the Shadow's fifth column can most likely be laid squarely at the feet of the Aes Sedai, who no doubt would strangle such groups in their cradles, FIGURATIVELY SPEAKING (lest I next be scolded for accusing Aes Sedai of infanticide), on the off-chance they might grow into another group like the Children whom the Tower cannot control. Just as Moiraine has to defend the ta'veren of charges of being like Artur Hawkwing, the Tower automatically is hostile to anything or anyone that might thwart their control, no matter how much good they do the world in the meantime.