Here’s your first plot twist: superglue was actually invented for war wounds. During the Vietnam War, medics would spray a medical-grade version (cyanoacrylate) on soldiers' gaping injuries to stop bleeding and seal them up until they got to a proper hospital. So, you’re not insane for thinking of it—you’re just late to a very old party.
But—and this is a massive “but”—the stuff in your junk drawer is not the same as the stuff in an ER. Hardware-store superglue has toxic solvents and stabilizers that can irritate tissue and cause a chemical burn or allergic reaction that makes your paper cut look like a vacation. It’s like using roofing tar to fix a china plate: it might hold, but the tableware won’t thank you.