To me, the phrase "win over" means "to convince someone to change their mind." It implies persuasion, not force. It is not the same as winning a game or winning a war. It implies persuasion, not force. This gentle meaning is also seen in
"He had a winning smile." (It makes you like him and smile back).
"He had winning ways." (Everybody likes him).
"At the beginning, everyone at the meeting was opposed to John's idea, but he was able to win them over."
I think it's perfectly OK to use it in the context of courtship:
"At first Elizabeth hated Mr. Darcy, but in the end he won her over."
A century ago, in books set in the U.S., you do read about men being called "lady-killers" and bragging of their "conquests." We do NOT speak that way any more.
There is a saying FROM ABOUT THE 1600s(!) which has made it into modern English as
"Faint heart never won fair lady."