By “kidnapping” the character, you remove her from the story, and by making her helpless, you remove her effect on events, even from afar. Without agency, the damseled character can’t show what she thinks or express herself through her behaviour throughout the game.
You can also use a person as a goal without forcing them to be as powerless. Where in the World is Carmen Sandiego is a classic example of this; Carmen has agency and purpose throughout the game. She travels the world and has adventures, and it’s up to you to track her down and bring her to justice. It’s a much more interesting game this way, rather than Where in The World is a Jail Cell Holding This Woman You Know.
Great article that gets bonus points for bringing up Carmen Sandiego in a feminist argument.