Specifically my question is about how we can get a new flu shot approved every year when we've learned how difficult the process is in getting the COVID vaccines approved for non-emergency use. The answer to this may be purely bureaucratic or it may be related to the fundamental differences in how the different vaccines work, so I wasn't sure if this was the right site to ask the question.
No, I don't think it's really a good fit for the Biology site. A similar question was asked (and I answered) at Medical Sciences.SE but has also been closed there:
Generally I think Medical Sciences would be a better place for questions involving regulatory approvals but they have strict requirements for prior research (more strict than even here at Biology).
If you want to learn more about the topic I'd recommend studying the flu vaccine specifically, because that's a rather special case among vaccines in that I can't think of any others that are reformulated on an annual time scale (possibly there are some veterinary vaccines or those not used in my country that I am therefore less familiar with).