No it wouldn't, because some people are concerned about the long-term effects.
Ok sure, I can understand that people have concernes over the unknown. Please let me try to appease your concerns.
We're primarily using the pfizer/moderna vaccines now which contain mRNA technology.
Messenger RNA was discovered in the 1960s, researched with cell delivery in the 1970s and underwent research to develop into a flu vaccination in the 1990s. a vaccine using mRNA technology was approved for human clinical trials in 2013 for a rabies vaccine.
The challenge of mRNA is how do you make the lipid nanoparticles that work as a exterior shell work with the mRNA which translates into the appropriates spike protein that triggers the immune response. The reason for the interim between testing and finally a working vaccine relies a lot on funding. When Covid hit the money available to find a working vaccine was near endless, and the bureaucratic process was speedlined and prioritized over absolutely everything else. A simple analogy would be to carry a full glass of water over a mountain, but you can't spill a drop. Normally you want to walk safe. Now with new shoes you could run at full tilt, but you still arrive at the other end with all your water in the glass. If its all there, the vaccine gets approved. Anything missing? Go back to the lake for a new glass.
No steps were cut, but the full weight of everything was available to them. Resources, finances, equipment and personel.
Side effects from vaccinations are indcredibly rare. (Worst case documented in 60 years is around 120 in 1 million or so if im not mistaken) Most people who develop any symptom at all after a vaccine see symptoms within the first two months, the vast majority of those again within the first few days. Covid-19 however has a list of potential long term side effects that incluce what we saw in patients with SARS: permanent lung scarring, Autoimmune conditions, chestpain, fatigue and tiredness/shortness of breath. I've seen interviews with 20 year olds in Norwegian newspapers who say they had Covid earlier in the year before they got vaccinated and now can't even get up a set of stairs without taking a break.
If we go back as far as the 1960s since the Polio vaccination program, there are no long term side effects from vaccinations. Not one.
mRNA has been used in trials for influenza, Zika virus, rabies and even cancer treatment. Its so incredibly well documented and tested.
Those who have concerns about the long term side effects of the covid vaccine generally don't have concerns based on medical data, but public opinion of those who are for one reason or another against getting vaccinated, period. For a reason that's summarized quickly as "Because I don't want to".
Its a ever moving goalpost that at the first stage of the vaccination program was moved to the corner flag at Old Trafford, but now it's used as a waypoint for rowers outside of Madagascar.