My off-the-cuff reading is that there are actually two types of characters that tend to be woobified: problematic types such as Kylo and Bucky, who commit in-universe atrocities and must be woobified in order to be sympathetic, and "cinnamon rolls", who are tortured by the narrative, circumstances, and/or fandom as a way to emphasize their sympathetic qualities. The latter is particularly common in moe anime, which is an entire genre devoted almost entirely to canon wobbie characters, or the "uke" (bottom) character in Boy's Love manga.