This seems like an urban legend

Absolutely not true. Wherever you read it is full of malarkey. I would go so far as to say it is impossible, since your be vomiting unless you were still drugged. It would take sedatives to keep you under long enough to swallow that much, and you can still vomit while sedated.

You also don't die from a full belly by itself.

Then, there's the fact that the stomach takes up some degree of water during digestion, and is breaking down any solids that it can break down along the way. You'd have to literally chug the blood to get enough in at once to distend the stomach, and no tonsillectomy produces that much blood.

Almost every single modern procedure uses some kind of cautery to stop bleeding, and the few that don't still take steps to do so.

Anyone, especially a small child, bleeding enough to die from swallowing it, would never be sent home. That's a sign of a major problem apart from the surgery.

And that's ignoring how much blood loss that would be. Even if swallowed, the amount needed to cause death wouldn't fill the stomach in a small child. Even in a bigger child, the stomach is bigger too, so you run into issues with realism there.

Tonsilectomies are done all around the world, and have been for ages. While complications can happen, this simply isn't one of them.

The poor child didn't understand that it's not good to swallow so much blood

It sounds like the swallowing wasn't the problem, the bleeding was. The swallowing just masked the true symptom, the bleeding, from being observed by others.