Where I'm from, in the 1960, the child mortality rate was ~225 per thousand. If you had 4 child, one was almost guaranteed to die before the age of 5. My grandparents had 7 children. 6 survived. It was the reality for literally every household.

Today the rate is ~25 per thousand. Look up the definition of vaccine in the dictionary and you'll know why.

Most vaccines do not wholly prevent the disease they vaccinate against. A few - like the rabies vaccine - do, mostly because they're slow-moving diseases that the body has plenty of time to mount a defense against, but many do not. In fact, the namesake of vaccines - the use of cowpox to vaccinate against smallpox - didn't confer immunity, just resistance.

Is the covid vaccine effective at reducing the chance of contracting, and the severity of cases of, covid? Yes. And that is what vaccines typically do. As for stopping its spread: the original vaccine came very close to herd immunity against Alpha, yes. What it didn't stop (but did mitigate) was Delta, which evaded some of the proteins it targeted and which was so much more contagious that even substantial protection wasn't enough for R << 1.

As for the word "vaccine": a vaccine is a deliberate exposure of the body to an infectious agent or a component of such in order to sensitize the immune system against it. The covid vaccine, which exposes the body to a protein that is a component of covid (an infectious agent) qualifies. The delivery method is novel, but the principle - using a component of an infectious agent to sensitize the immune system - is the same.