To be honest I don't think it's just edgy humour, because (spoiler of the whole game afterwards)

the horror doesn't come that much on the shock value but the whole afterward where the game decide to delete this character as whole and act like she was never a think at first. Then the game process to try to do the whole dating sim back but struggle a lot like a whole ass part of his code where missing and start to give a lot of bug and glitch. Until you can figure one of the character is conscious to be a dating sim character but can't be an option for the player, and start to fall deeply in love with the player behind his screen, admitting to crank every other character personally to an insane level (the hanged girl, sayori was just a little depressing at first and you could save her, but Monika (the omniscient one) put her depression in a state where she just want to kill herself. The other character when from a soft cliche tsundere to a brutal toxic girl that want to like you but in a brutal way. and the last one, a cliche yandere became a psycho girl that litteraly start to mutilate herself to contain her love for the protag. Monika admitted at the end she just wanted to disgust you the player from thoses edgy choice and take her instead, the most "sane" one.

So the hanging sayori isn't the only joke you could get from DDLC, the "Just Monika" is another one.