There's an extremely long list of creative works made to be satire that idiots don't understand. The Colbert Report, Starship Troopers, hell there are people that think Homelander is the good guy in The Boys. I think it's a little too restrictive if someone can't make a point that certain jokes are lame out of fear of how a few idiots might misinterpret it (many times deliberately). There are people right now arguing that the Barbie movie is actually "anti-woke".

Also consider how many people actually did get the message. They heard how lame the jokes sounded and realized how stupid fat jokes are. From what I recall, the game really makes an effort to deconstruct these jokes. There is no audience within the game laughing at the jokes, and the evil AI explicitly states that it's making the jokes solely to to negatively affect the protagonists psychological well being to prevent her from achieving anything. And the jokes are directed at the player, so someone that might have made these jokes themselves hears someone else directing these jokes at them and can hear how lame they are. The evil AI comes across as lame, petty, and desperate. Someone who made fat jokes before playing that game might have had the realization "is that how I sound?"

I feel like the people that didn't get the context likely made fat jokes before playing the game, and would be making fat jokes even if they never played the game. So I don't think there was any negative impact. Someone that committed to being an idiot isn't going change in either direction from any kind of media.