"The results of the investigations showed that RotaShield® vaccine caused intussusception in some healthy infants younger than 12 months of age who normally would be at low risk for this condition. The risk of intussusception increased 20 to 30 times over the expected risk for children of this age group within 2 weeks following the first dose of RotaShield® vaccine."
See http://www.cdc.gov/vaccines/vpd-vac/rotavirus/vac-rotashield.... Now, that's not much of an increase in absolute terms because intussusception is relatively rare, but so is infants dying due to rotavirus infection in the US - the estimates I'm seeing are 20-60 deaths a year without vaccination. Even a relatively rare adverse reaction is enough to outweigh the benefits of the vaccine in the US.
Now, you're right that the clinical trials "detected no statistically significant serious adverse effects". That's because they were so small that, even though the adverse reactions were common enough to outweigh the benefits, they couldn't actually detect them as being statistically significant. Hell, even if the vaccine was somehow hypothetically killing ten times as many babies as would've died from rotavirus, I don't think the trials used to approve the vaccine could have detected that. That's kind of worrying.
2015 - 1999 = 16
The point which you are overlooking is that even something as tested as vaccines still had an undiscovered problem. So you are being overconfident.
You want to say the risk is low enough that it's worth it? Fine. But do not be so quick to dismiss concerns.
PS your math for 40 cases a year seems incorrect, there are considerably more than half a million babies born each year.
https://en.wikipedia.org/wiki/Rotavirus_vaccine "In 1998, a rotavirus vaccine (RotaShield, by Wyeth) was licensed for use in the United States. Clinical trials in the United States, Finland, and Venezuela had found it to be 80 to 100% effective at preventing severe diarrhea caused by rotavirus A, and researchers had detected no statistically significant serious adverse effects. The manufacturer of the vaccine, however, withdrew it from the market in 1999, after it was discovered that the vaccine may have contributed to an increased risk for intussusception, or bowel obstruction, in one of every 12,000 vaccinated infants."
Note the may. The number is so low as to make it difficult to correlate. That's 40-50 cases in a year. Aspirin kills that many in a year and we don't consider it unsafe and we take it for things which are lots less pressing.
However, they pulled the vaccine because the perception in the United States is that we have adequate treatment for Rotavirus without the vaccine and it simply wasn't worth trying to correlate.
"Meanwhile, other countries such as Brazil and Mexico undertook their own independent epidemiological studies which demonstrated that 4 deaths were attributable to vaccine, while it had prevented approximately 80,000 hospitalization and 1300 deaths from diarrhea each year in their countries."
4. You will get that many people dying of an allergic reaction to anything if you give it to several million people.
More people die of peanuts in a year. Is that "unsafe"?