Why is the radical of 难 > 隹 and not 又?
As @50-3 has mentioned, the 难 is the simplification of the traditional character 難. Most Chinese characters are phono-semantic compounds (形声字), in which the radical hints at the meaning while the phonetic hints at the pronunciation. In the case of 難, the phonetic component is 堇 while the radical is 隹. In modern Chinese, the pronunciation of 堇 has diverged from 難, but their pronunciations in Old Chinese were presumably closer.

Why is the radical 隹, which typically refers to birds? This link gives the explanation: http://font.zenque.com/zidian/4796.html


  難，今单用简作难，读nán，本义指一种鸟。因其从“堇”（火焚人祭雨）取得声义，故借以表示①困难；艰难


難 originally referred to a type of bird, but because the pronunciations coincided, the character ended up being borrowed to also refer to hardship and difficulty.