Vaccines work to decrease infection &/or severity of infection. All vaccines have failure rates, dependent on the immune status of the patient and the mutations of the virus or bacteria.
Half of the Covid vaccines types used in the US (Moderna & Pfizer) are MRNA vaccines that stimulate production of an antigen protein for a short period, certainly not in every cell in the body, nor do they cause the disease or induce even a mild form of the disease – as the measles, smallpox, and the oral polio vaccines do.
Most current vaccines use either an attenuated virus, a killed virus, or an antigen protein produced by recombinant DNA in bacteria or yeast. These last have never been part of a virus or bacteria. And, in fact, recombinant DNA is used to produce the human insulin to treat diabetes.
The mRNA in the Maderna & Pfizer Covid vaccines aren’t continously replicated by the vaccinated person and isn’t incorporated into the DNA. They are present in the human body about 2-3 days, mostly in the local muscle tissue and lymph system, with some in the spleen, more rarely in the liver.
Yes, you more than likely had an attenuated form of those infections when you were vaccinated. Current measles vaccines can cause infection that can be spread to immune compromised contacts. One way polio was all but eradicated was because people who came into contact with the babies who received the oral vaccine feces would also be infected – effectively receiving a booster unknowingly. We stopped using the oral polio vaccine in the 90s because 1 in 20 million children got a polio like disease after the 1st ( only the 1st) dose. We warn families to avoid contact with immune compromised people for a time after a child gets the measles vaccine.
The process was used years ago in the vaccine against ebola, so there’s history for use in humans.