To provide a simper, clearer answer: It's not necessarily the sprouts, but the coloration of the potato. If you go to peel it and it's green, don't eat it. When it turns green it basically is producing solanine which is toxic if consumed in large amounts, causes digestive distress. Opinions are mixed. Some say throw it, others say just peel until white. So overall sprouts don't matter, it's what is underneath the skin of the potato.

Possibly I am an idiot here, as a whole lot of people seem to think its ok to cut out the sprouts and use them anyway?

I was raised that basically once they start sprouting, at that point you should be planting them, not eating them.

https://www.healthline.com/nutrition/sprouted-potato

As a potato sprouts, its glycoalkaloid content begins to rise. Therefore, eating potatoes that have sprouted can cause you to ingest excessive amounts of these compounds. Symptoms typically appear within a few hours to up to 1 day after eating the sprouted potatoes.

At lower doses, excess glycoalkaloid consumption typically leads to vomiting, diarrhea, and abdominal pain. When consumed in larger amounts, they can cause low blood pressure, a rapid pulse, fever, headaches, confusion, and in some cases, even death (1, 2).

I'm not a doctor or dietician or chemist or potatologist and I have no idea if this is BS, and it seems like there is fairly widespread ... mixed opinions on this, but these here doctors say eating sprouting potatoes can make you sick, even if you cut out the sprouts.

After they sprout, they also produce poisonous chemicals