I am sorry I don't mean to offend you but what you just wrote is complete non-sense to me. Stephen Jay Gould probably could have improved his work had he lived long enough to see what theoretical physicists have been working on today.
The thought that "correlation implies cause" is an invalid assumption? Yes correlation absolutely and positively must imply cause and simply that. Without correlation we have no real reason to invest into discovery. Are you going to waste valuable time researching a topic that is not based upon some correlation? Without patterns there are no real problems just unaccounted for randomness.
Early on in the debate, I would have agreed with your statement that improved diagnosis and detection have adversely affected the statistical analysis. However, based on just common sense alone, we are clearly far past that point now. There are simply too many cases of parents experiencing the same issues with their children and vaccination. It is important to note here that the majority of vaccination is safe. Statistics can prove that as well. Here in the USA we should be questioning the number a vaccines required, the age when they are administered, their effective timeline, and one vaccine in particular the MMR.
I would love to hear your comments on "many other variables". I see this as a weak argument. It is the job of the scientific community to set up parameters during research which account for such variables. Research must be conducted properly no matter how difficult. You must keep pushing forward until you discover a way to conduct it properly.
One more thing to note, you choose to analyze just one research paper. There are many more out there with similar findings. Just because you found one with bogus research doesn't mean that the problem doesn't really exist. You kinda cherry picked it. I would like to see you find another paper on the same topic that you cannot dispel the research. One that in your own opinion shows that they did the research properly.