The reasons for assessing covid vaccines to be safe divide into (a) those that do relate to other vaccines, and (b) those that were specific to the covid vaccine testing program.
In (a), we have things like "we know how vaccines work because we've administered literally billions of doses and amassed vast knowledge about human immunology" and "we know how statistical power works in giving us good insights into treatment safety and efficacy" and "vaccine side effects show up quickly, consistent with their pharmacodynamics" and "endpoint accrual depends on event rates and sample size".
In (b), we have "statistical power means that larger trials give faster and more accurate insights into safety and efficacy, and covid vaccine trials were about 20 times larger than a standard trial for a pharmaceutical" and "recruitment and dropouts are the bane of standard trials, but for the covid vaccines there were enormous numbers of people volunteering so recruitment including followup recruitment for dropouts was much faster" and "high community incidence drove very high event rates, meaning endpoint accrual was much faster than for a standard trial". Endpoint accrual scales about linearly with sample size, so the covid trials reached the required endpoint count 20x faster than a standard trial. The net effect of this and all those other factors like recruitment and community incidence was that the trials reached statistical power in three to four months vs two to three years. Standard errors shrink by sqrt 20, ie about 4.5, so confidence intervals are 4.5x narrower for the same observed rates, and effect estimates are 4.5x more precise. Rare safety event detection improves about 10 fold, ie you can spot 1 in 10k rarity events with a 40k person trial where a 2k trial could only spot 1 in 1k rarity events. Etc.
Finally, we need to consider that covid vaccines were rolled out on the scale of billions of doses, meaning that we have giant quantities of data from vast numbers of countries and populations, giving us extraordinary post-launch insights into the vaccines' safety and efficacy. It's not like a CF drug which is taken by a handful of people, and where rare events can remain undetectable.
We have *hugely* better insight into covid vaccine safety and efficacy than we normally get for a treatment, because of all of the above.
As always with this sort of story, whether it's "vaccines aren't properly tested" or which groups are most likely to abuse children etc, the story is the inversion of the truth.