Armor is not the end all decider of all combat. In Africa and the middle east, many warriors didn't wear armor at all, either because it was too hot. And you don't need to be collapsing from exhaustion during combat. In ancient times many people didn't wear it because they didn't know how to manufacture anything beyond light leather armor. In Western Europe, the vast majority of soldiers didn't wear armor at all. When it is talked about chain mail and plate mail, those things are highly expensive and so are pretty much restricted only to Knights that constituted the aristocracy. The Mongols were nomadic. They're not going to be manufacturing any kind of elaborate armor and on top of that, their style of cavalry warfare was very different than that of Europe, and wearing plate mail isn't conducive to a strategy involve light/highly mobile cavalry archers. And the defeat of Western forces isn't necessarily indicative of the failure of armor since the peasants conscripted by a lord to fight are generally going to be unarmored, untrained, and fighting with improvised or very cheap weapons.