Recently, her family advised me to be patient and try to do things to gently win her back. They suggested texting her from time to time, initiating conversation. Basically, since I am the “sane” one and also since I am a man, I should be the one apologizing first and try to gently win her back. “Women like to be charmed and treated like a princess” they said.
Sounds like persuading and chasing to me, which to my understanding is something I shouldn’t do when detaching..? Or should I try it and see what happens?
I think this is terrible advice. If she's been talked into marrying you, or fell in love with the idea of marriage, then she's not an emotionally mature adult woman. If she's described by her own parents as childish and selfish, then she has probably got way, in part, because they've treated her like a little princess instead of expecting her to take responsibility for her own decisions and the consequences of them, as is appropriate for adults. The fact you're both staying with your parents instead of having your own places, or arranging your own alternative living spaces if it is needed is another sign emotional immaturity is involved.
The mature, adult and respectful thing to do is to respect what she says - she doesn't want to be married. You don't need to 'let her go' - she's gone. What she does here on out, including seeking a divorce - is down to her and within her power. You get on with GAL, working on yourself and your own maturity, and you leave her be. A mature man doesn't find a pouting woman who needs to be cajoled into a relationship remotely attractive.