"If they were as safe as other treatments they wouldn't need a blanket liability immunity." Citation very much needed for this inference.

Even if I granted every single paper's premise here. I'd still much rather have a living child with a slightly higher chance of allergies or asthma or <insert survivable condition here> than a dead child. How quickly we forget how bad things once were. Do you dispute that vaccines also accounted for 40% of the decline in infant mortality over the last 50 years? And before that, TB, Flu, and Smallpox killed uncountably many people. Vaccines are a public good and one of the best things we've ever created as a species.

Do you also have theories about autism you'd like to share with the class?

A very good point. These studies should be comparing QALYs (quality-adjusted life years, a measure of disease burden) instead of relative prevalence of a handful of negative outcomes, the latter of which is much more vulnerable to p-hacking.