You have to be very careful with this kind of informal statements, because probability is a tricky thing.
For example, assume that a vaccine has a probability P of causing long-term bad effects and that we observe historically that 1000 vaccines have no long-term bad effects. What can we rationally say about P? It would be tempting to conclude that P is zero or at least negligibly small (maybe 2^(-1000) or something). However, this cannot be the correct answer because if P were 1/1000, you would observe the historical pattern with probability .999^1000 ~= 1/e, or about 1/3 of the time. A more sophisticated analysis would conclude that it is very likely that P is somewhere between 0 and 2/1000, and if you believe a thing called the "Laplace Rule of Succession", the expected value of P is something like 1/1002.
Now, 1000 widely deployed vaccines sounds like a reasonable upper bound to the number of vaccines ever produced, which would suggest that the probability of this vaccine being safe is probably 100% but the error bar is O(1/1000). O(1/1000) is about the same as the probability of the virus killing me, so it is perfectly rational to have doubts.