Babies are not canonically delivered by storks, it's been established that babies are born in multiple ways in Mario. Yoshis hatch from eggs, Lumas are born from stardust, Rosalina had two biological parents and a brother, the DK family are biologically related, etc.
That whole plot point of Peach potentially being Bowser Jr's mother was completely non-sensical and unnecessary from the get go. If you want to assume storks are the only way a baby can be delivered to two loving parents in Mario, that implies that Peach and Bowser are indeed in love, which introduces the theme of cheating on someone (Mario, as yes, they are actually a couple despite some fans trying to grasp onto that "they're just friends") in a Mario game. If, like pretty much everyone else, you assumed that babies in Mario could be born through sexual reproduction, then you run into the issue of Peach and Bowser having an implied sexual affair. Oh, and the idea of sexual attraction has been lightly addressed in the Mario universe, so it's not like people are weird for jumping to this conclusion. Peach in particular has been used for sex appeal by Nintendo on numerous occasions too, from commercials to how she's portrayed in Smash and some of the Mario spin-offs.
The reason why this whole idea was stupid is because, regardless of how you interpret baby birth in Mario, they are still going dangerously close to bringing up adult themes in a Mario game, to the point where they can do literally nothing with this plot thread other than cop out of it. Which they did, but since they knew they would cop out anyway, why did they bring it up? It's an idea that's literally dead on arrival. It adds literally nothing to the narrative.
This isn't comparable at all to the ice cream scene. Regardless of whether or not you think it's cringeworthy, it still makes sense and adds something to the narrative (i.e. highlighting Roxas' innocence and limited understanding of friendship that was based off when Axel literally said "Friendship is eating ice cream together and laughing without a care in the world", as well as just how much he valued Xion). And while you may be completely cold to the scene, most people literally view this scene as the most emotional, tear-jerking scene in the entire series and perfectly comprehend the existence of that line and see how it adds to the narrative, and it can't be the worst thing ever if that many people connected with it. It has a clear purpose, the symbolism is obvious, and for many people, achieved the emotional impact it was going for. There's nothing stupid or difficult to comprehend about that, nor does it unintentionally bring up themes that are inappropriate for the game's demographic.
Also that manchild comment is just uncalled for since even forgetting the point about him only existing for a year and starting as an empty vessel, Sora (whom he is based off of), is still only 14 years old at this point. The death of a close friend is still a difficult, messed up thing to experience for someone of that age. I could still very much see something like that coming from a regular person in that scenario. That's why so many people connect with the scene.
KH has problems with certain story elements, it's not perfect (no good thing is though), but the ice cream line is as far from that territory as you can get.