That story never made historical sense. Germ theory was just beginning to be accepted by the medical establishment at the turn of the 20th century. Doctors still did minor surgery without antisepsis. The idea that invisible animals (bacteria) could kill a healthy man would have been considered ridiculous by tough Indian fighters and soldiers of the 19th century. If you told them that blankets could kill Indians, they would have laughed in your face.
That's somewhat untrue. You don't need germ theory to grasp the concept of contagion, and there are primary sources where the idea of using possessions of smallpox victims to spread contagion was suggested. It's pretty dubious that it was ever carried out or would have been effective (hence the myth of smallpox blankets), but it's not as if people were stupid and unable to grasp that disease was somehow associated with the sick or their possessions.
People used to throw bodies into wells and over castle walls during warfare because they understood contagions existed. They simply didn't understand the mechanism.
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