As an aside. The rare narcolepsy side effect that was associated with the swine flu vaccine was found to be associated with the adjuvant (booster) used in it, not the vaccine itself, and has been withdrawn. The Pfizer vaccine (and, I believe, the Oxford one) don't contain an adjuvant as create a strong enough immune response on their own.
The reason that people are splitting hairs over "long term" effect from a vaccine is that is not really how they work. As a PP said, they are designed to go in, stimulate your immune system to create antibodies and then are dispersed (within a matter of days). Your own immune system creates the antibodies to the disease that that vaccine was mimicking. So any undesired negative side effects are going to be either immediate in an allergic reaction to one of the components - would happen very quickly and, likely, only to people who have an "abnormal" highly reactive immune system that is easily overstimulated (what an allergy is) and would probably know that they have multiple allergies, carry an epipen etc - or, a consequence of your own body's immune system doing something else undesired/negative as a result of the "kick" it received from the vaccine - what happened to the boy that developed narcolepsy and an auto-immune condition after having the swine flu vaccine. He most probably had an "odd" genetic outlier immune system to start with and would most likely have had auto immune issues, of some degree, as he got older, with or without the vaccine.
So the question for me has always been along the lines of, is the chance of me or my children, getting this disease, and could it be bad enough, much higher than the chance of me being a genetic outlier with an "odd" immune system that may respond adversely to this vaccine. So far, the answer has always been in favour of the vaccine, except for chicken pox where I decided that there was no need for the vaccine and I was ok with my kids just getting chicken pox.
As an adult of nearly 50, I think I can be reasonably certain that any unusual immune response would probably (not definitely, it's all about probability) have shown up by now. Just as I know I can drink alcohol in moderation without any real danger of becoming an alcoholic. Yet there are plenty of people out there who shouldn't really put alcohol in their bodies because it makes them violent, they get addicted, become alcoholics etc.
Obviously with children you probably have to be more careful as their immune system / genetic outlier status is less tested. Which is why they don't give vaccines to under 16s if they are not particularly affected by the disease - like here with Covid. Having said that, my children have had all their childhood vaccines and have the nasal flu vaccine every year, with zero side effects, so the chance are in favour of the fact that they would be fine with getting this, or any other, vaccine.
It's a simple equation for me. How bad is the thing they are vaccinating against and how much do I not want to get it (no health concerns so would likely be mild, but I worry about long covid effects having seen some friends suffering badly, could pass it to overweight, diabetic mother and kill her, is stopping me socialising and having fun) > the chance of me suffering a severe unanticipated side effect from the vaccine itself (very low for me, no previous auto-immune issues, no allergies, had lots of vaccines in the past without issue, it's a simple formulation vaccine with no weird adjuvants or chemical preservatives that I can see).
Scrutiny is entirely appropriate and everyone has to weigh up that formula differently. If you're pregnant, carry an epipen, have an auto-immune condition etc, the equation may be different than mine. I do find it weird though that people don't apply the same degree of scrutiny to trying a new food, or a new brand of Gin, or a new medication that the GP prescribes for blood pressure, or whatever.