I think the overarching rationale behind a vaccine is that if a high enough percentage of the population get a vaccine(that presumably grants high levels of immunity, that may or may not be the case in COVID, more data needs to be seen), that the disease will no longer be able to spread since its potential host population is immune. This is all to protect individuals, who due to old age, compromised immune systems, or other malignancies are unable to get the vaccine,. Flu is an excellent example of this, young individuals are less likely to die from the flu and are also have stronger immune systems and can handle a vaccine, elderly individuals with weaker immune systems can not get the vaccine but are at greater risk, hence younger individuals should get the vaccine to prevent potential spread to older people. Its altruism on a larger scale.

also keep in mind that 450 people did get Guillain-Barré syndrome with the 1976 vaccine campaign, but that was out of 45 MILLION People vaccinated. 1 case per 100,000 vaccinated individuals, and this vaccine is widely regarded as a debacle and outlier in terms of risk. Whether or not the Ford administrations panic over the new strain of flu was rationale(it probably wasnt) the risks we are talking about are minor compared to the death rates for flu or in particular COVID.

this is not to say healthy skepticism for science is a bad thing, but perspective and altruism for the broader population are key.