I've read that most Europeans (thanks to having plagues) were immune to many diseases they brought over to the new world. If they couldn't, or didn't, catch the disease how did they know the disease was in the blanket? What proof did they have that anything was in the blanket? Did they just take an article off of a victim they had on the ship; which just happened to be a blanket?
You seem to be assuming the blankets were given with the intent of spreading smallpox. That is not the case. If you're immune to whatever is on a diseased blanket, it just seems like a normal blanket to you.