What kind of cheating do you want to prevent by teaching kids cursive? Them using AI to write essays or reports? But even before AI kids would just download essays off the internet, and even if you didn't have internet, you could buy a book with pre-written essays at a bookstore.
It worked quite well, since the topics mostly stayed the same and require you to analyze a work of literature(e.g. "Juxtaposition of Raskolnikov and Svidrigailov in Crime and Punishment") and teachers didn't use a plagiarism check cause they weren't technologically savvy enough, and even if they were, back then OCR wasn't that good even for printed text, I assume recognizing cursive would've been practically impossible.
So yeah, I don't think cursive is gonna help deal with cheating.