The question that you're asking is a good one. I don't think that there is an easy answer. Let me ask the question that you're asking more broadly, "what is the burden of proof to say that ANY medical intervention is responsibility for a side effect or allergy?" The truth is that causation is difficult to prove with 100% certainty. This is also true of medical diagnoses as well, for that matter. For example, a patient is prescribed medication X and within days develops signs/symptoms Y. Is it causation? It really depends. If medication X is highly linked to signs/symptoms Y, then it increases the probability. If medication X was introduced immediately before the patient developed signs/symptoms Y, and there is not a better way to explain why the patient developed those signs/symptoms, then that would also increase the probability. But the probability is never 100%, and we have to use our best clinical judgement. Another example, patient develops signs/symptoms Y and it fits the clinical pattern of diagnosis Z, and that is supported with appropriate testing that also suggests diagnosis Z, then that would increase the probability that diagnosis Z is the correct diagnosis. A diagnosis will be made but there is never 100% certainty. We use probabilities to determine the MOST LIKELY explanation for something happening, but there are no certainties.
Regarding vaccines, and I'll review this a more broadly because I think that it'll help you understand the challenges of studying the topic. Here's a case: a baby develops profound nausea/vomiting that necessitates hospitalization. The baby is a twin and the other baby is completely fine. The sick baby gets resuscitated with fluids and monitored in the hospital for a few days, as its kidney function was off (likely because of dehydration). The baby would have likely died if it wasn't for medical intervention. Tests are done for an underlying pathogen, and there is nothing found (which makes since as the baby's close contact twin was completely healthy), and there was otherwise no other evidence of disease. The baby does well enough to get discharged from the hospitalization and is back to normal after a week or so without any future problems. Oh yeah, might not be important but the sick baby did get the Rotavirus vaccine the day before getting hospitalized. There is no "rotavirus vaccine" side effect lab work or imaging to confirm the diagnosis. So did the vaccine have anything to do with the hospitalization. My clinical impression was that there was a very strong time correlate, Rotavirus is a live vaccine and has been linked to rotavirus-like reactions including nausea/vomiting/dehydration, the other twin never got sick, and there was nothing else to better explain the hospitalization, so though there is no PROOF, a vaccine related side effect is the most likely explanation. My wife and I are both in medicine, we aren't dummies...but we had a PICU doctor try to convince us that the reason why our daughter was hospitalized wasn't because of the vaccine, because the doctor was afraid that it would scare us away from future vaccinations.
Here is another example: a 22 year old ends up being found dead by his roommate. On the scene it appeared that he had a seizure. Pathology ends up showing that the young man developed a condition called disseminated intravascular coagulation (DIC), which is an unusual presentation of abnormal clotting in some parts of the body depleting clotting factors and causing bleeding in others. It was believed that this condition resulted in the seizure/stroke and death. He was previously completely healthy without any family history of anything remotely concerning of a bleeding/clotting disorder. The young man had a vaccine less than two days before the event. Unfortunately, the vaccine is relatively new and therefore there is not enough safe data to be powered to capture an event so rare. But could this even have occurred randomly, unprovoked? ANYTHING is possible, but I'm going to share with you why I think that this particular case is concerning. DIC is so rare that is almost exclusively found in patients in the ICU. There was a study performed at Mayo Clinic that showed that the incident of DIC in the ICU was about 1 in 200,000. Understand that Mayo Clinic has arguably the sickest patients in the country. Truth be told, many patients that are in ICUs in community centers would be managed in either general floors or step down units at Mayo Clinic because the average level of complexity is so impossibly high. So that 1 in 200,000 would likely be EVEN LOWER in a community ICU. And we are talking about the ICU, not the general healthy population. The incidence of DIC in people walking the streets has to be 1 in the tens of millions, statistically. It's essentially impossible. So when a medical intervention occurs within two days of this impossibly rare disease resulting in death occurs and there is no better explanation, that catches my eye. Did the COVID vaccine cause the death? There is no COVID vaccine side effect blood work or imaging to confirm diagnosis, you just have your clinical impression based on probabilities.
I've had patients develop all kinds of weird reactions to treatments that I've administered. Many of which aren't even recognized side effects, or are incredibly rare. But do I go out of my way to defend my treatment and disregard sound clinical judgement? Heck no. Unless I have a good reason to otherwise explain the reaction, I'm labeling the intervention as a side effect or allergy and not doing it again. But for some reason, rationale decision making has, in many ways, completely eluded medical providers on the topic of vaccines.