Yes, I think that your question is very pertinent, and it demonstrates the limitations of post-marketing surveillance.
On the early part of your question about how we establish a link, a causal link, between a marketed product, a vaccine or a drug, and the adverse events that we see, the answer is that it is with extreme difficulty and a great deal of time and effort. Once the drug or vaccine is out on the market, it becomes almost impossible to do the randomized control trials that are done prior to marketing and that should be done in a manner that is sufficient to identify problems. Once the drug or vaccine is on the market, randomized trials are not really an option, so the ability to demonstrate a causal link is lost. You can demonstrate an association. This is a scientific term that means on a balance of probabilities this is a causal problem.
The second issue is that once a drug becomes politicized, the science behind the drug and the surveillance done on the drug or vaccine becomes secondary. There is evidence both in Canada and in the U.S. that this particular product became the subject of tremendous political pressure, to the point that the company's own top researcher for the clinical trials spoke out publicly to say that both the company and the regulator were taking action to provide the vaccine to very young girls--as young as nine, 10, and 12 years old--when in fact none of the studies was done in that population and there is no evidence that the drug will be helpful or what harms might arise.
Lastly, your point about informed consent is crucial. As a parent, I've had my daughter announce in grade school that she was getting a vaccine the next day. When I asked, what have they told you? She gave me a little pamphlet that had nothing about the vaccine, just a public health message that it was to be given, and she, at the age of 12 or 13, was going to sign for her own vaccination. It was a vaccine that I did not particularly feel confident about.
You're absolutely right that we are taking tremendous liberties with people's health in a manner that would be offensive to the average citizen if they understood what is really going on.