I don't know, I think people are pretty well aware of the yellow card scheme but (as zeph said) if they are told at the first stop GP that it's coincidental then they have to be quite determined to report it, especially when there is little to be gained.
I think you are much too optimistic and naive, glib even, to say "it'll all even out over time". Two decades of MMR and reports of adverse reactions are still not taken seriously. Even though Cochrane said the inital testing was largely inadequate. Perhaps it won't ever be.
I'm repeating poface in saying too much evidence is dismissed as anecdotal. I don't say proof, it's not proof, but it is evidence.
Instead of further research someone saying hmm this really should be looked at millions of pounds have been spent on promotion campaigns and worthless retrospective epidemiological studies to cloud the issue.
If only there could be a true study comparing vaccinated and unvaccinated populations. In the study I mentioned earlier, but didn't link, it said the highest rate of CRS is found in the Amish population. The answer seems/is obvious -- it's a self-selecting, non-vaccinating population.
However I can't imagine that those who are very keen on the idea of vaccinating would be awfully keen to make or draw attention to large scale comparisons with Amish communities.