We have to be careful when comparing the raw numbers. The AstraZeneca vaccine is a vector vaccine and part of it's expected mechanism of action is triggering an (unpleasant) immune response. I'm not sure where the official definition draws the line between side effect and just normal unpleasantness, but I've heard that most people experience symptoms like a mild flu, and pain at the injection site. OTOH I've heard anecdotically that the Pfizer/Biontech vaccine has much less reactions.
Anyway I think we should not lump the mild/expected side effects and the serious and life threatening ones together. Unfortunately it's quite difficult to find reliable numbers for probability of e.g. blood clots after vaccine compared to the general population.
As a physicist, I'd also like to see the probability in the general population, but this population reweighted to match the population that has been given the vaccine. (E.g. if 0.1% of the population are nurses and 2% of the vaccinated are, then multiply their frequency by 20). When I just see a number, I never know how much they take this kind of confounding factors into account.