A 'perfect' vaccine does work like that, where the specific entry-point the pathogen uses gets patched and completely prevents infection. Very few vaccines work like that unfortunately.

The vaccine that was delivered was instead a 'communication blitz and drill' to train the body 'if you see this guy kill him real good!' where it used mrna to directly put COVID onto the immune system's priority kill list and supplies some target practice.

The problem with the second type is the immune system can't be everywhere all at once and so some amount of infection does occur which does result in those 'side effect of the infection' occuring. In addition that 'target practice' can and does cause similar side effects.

All that said, imagine how much worse those side effects would be if the virus was allowed to infect more in addition to hangig around longer!