Spoken like a true ignoramus.
Look, like everyone else I haven't seen the game, so standard disclaimers apply.
But: Context matters. There's not exactly an epidemic of the subjugation of men by women in the world. Whereas there are large numbers of people even in the US who think women should be subservient to men (hell, it says so in the Bible!) and generally treat women more like property than like people.
The "damsel in distress" canard is about as deeply ingrained a theme as exists in entertainment, and its prevalence is indicative of widespread sexism to begin with (not to say that each instance of it is sexist, but for girls who play video games, it's still a novelty to see a female character who doesn't have to be rescued by a man... And extremely rare to see a female character actually rescuing a male.)
That said, on it's surface, this sounds much nastier than a damsel in distress. Usually the damsel at least is assumed to be an actual person whose freedom is in itself worth fighting for. "Capture the Babe", though? First, this is a "take the other guy's sexy chick" thing, which grants zero agency to the woman - whoever takes her gets to have her. Which is already pretty typically misogynist even before you get to the hysterics and the slapping, which are respectively a negative stereotype and a tacit acceptance of violence against a woman who I presume doesn't fight back.
And to the people calling it "satire"? Satire condemns its target, usually by presenting an exaggeration of that target. Unless the audience of primarily young men is expected to take this game as a satirical commentary on how accepted various sexist constructs are in video games, I don't see how you look at this and think "Satire."
Do I think it's going to make people act violently toward women? No, but it will reinforce some negative attitudes toward women that are already prevalent in society, and any girls who are playing the game will have just another example of how people like them are just there to be claimed by the Big Men.