There are a few problems I have with your analysis of the fighting. First of all, the guide refers to the fighting between the survivors of the Light and Shadow. This was when Latra Posae won her acclaim - if she is the pre-incarnation of Egwene, I'm sure it was by taking credit for the work others did to combat the surviving Shadowspawn, Darkfriends and Forsaken, as well as protect people from the depredations of bandits and the like. In any event, there was a LOT for Aes Sedai to do in those days. RJ said in a QotW answer that in some places during the War of the Shadow they were down to swords and axes and horses levels of tech from the damage to the infrastructure. With all this in mind, people would have been trying to get things up and running and in the first place, might not have noticed the insanity or realized the odd madman they encountered was part of a universal epidemic. So with all that was going on, it probably took a while to catch up the male channeler problem. Thus, it would have had time to pick up momentum and become all but insurmountable, rather than sane and well-equipped female Aes Sedai having the advantage over their brothers. It more likely caught a lot of them by surprise, and did even more damage to the surviving sisters before they started working with a plan or a strategy to deal with the crisis.

Secondly, they did not grasp the entirety of the problem - for those sisters, it was not as simple as round 'em up and gentle or kill them. Initially, they tried to help their one-time brothers (and don't forget there were still-sane males on their side, which would have impeded any serious consideration of a such a drastic solution). They would have first tried reasoning with the madmen or subduing them without causing harm (not least because they'd want to try to study the taint and see what they were dealing with). Such efforts would have led to reduced success on the part of the Aes Sedai. Look at Verin during the Cleansing - she got greedy and tried to capture Graendal, hungry for the knowledge and other advantages to be gained from such a captive, and as a result, her initial strike failed. She did not take into account how difficult Graendal would be to shield with an angreal. Had she blasted the Forsaken with all she had, Graendal would not have had the innate and passive defense that overwhelming strength is to a shield - even if she did not die, it would have been because she would have had to take measures to counter Verin's attack, and thus surrendered the initiative right back, which might have saved Verin's fellow Brown. This scenario no doubt played itself out many many times in the years after the Sealing, with sisters trying and failing to corral their insane brethren only to see even more woe when their initial efforts proved insufficient.

Third is the fact that the madness seems to take an invariably malicious nature, as to be expected from something caused by the Dark One. Both of the contemporary madmen we see support this. One is inspired to start tearing apart a building for materials to protect his charge assigned right before he slipped, and the other sees non-existent Myrdraal everywhere - lashing out at them would have been the next step. Thus the reflexive view of madmen as incompetent and easy victims is probably also flawed. In their minds they had good and specific reasons for the various acts of destruction they wrought, they were not simply flailing around at random. They would have been driven to destroy and harm as many as possible, which probably made it even harder for the sisters to track or catch them. I would bet that not a lot of them wandered into open manholes or blew themselves up without harming anyone else.

For reasons like these, I strongly doubt they went down as easily as you suggest or that their immediate successors without training were as inept and relatively harmless as the typical untrained channeler. As I said, this madness is not mundane mental incompetence or errors in judgement, it is a malicious distortion of rational perceptions by the Dark One's touch. It would not surprise me at all for a guy who went through his early adulthood only unconsciously touching the Source when he needed something direly, until he went mad, to suddenly consciously, deliberately and maliciously use the Power to much greater effect.

On the other hand, what I do have a problem with is the fact of human survival in the extremes of geographical upheaval and climate fluctuations. "Where the food" comes from is a frequently neglected detail in WoT, despite the lip service paid in the main characters' immediate storylines. Background stuff like where the Tinkers get theirs (their studious avoidance of settled people argues against obtaining it all through trade, and IIRC they don't eat meat, so it's not as if their food accompanies them on the hoof) tends to be neglected. In this case, the description of the rapid and extreme changes argues against any kind of serious efforts at agriculture, not to mention mass extinctions as niche animals struggle to adapt to new climates, while lacking the rational faculties to adjust. If issues like that can be surmounted, the supply of male channelers is relatively minor.