You vaccinate yourself to protect yourself. People who are vaccinated for a disease can and do spread said disease. It's this way for literally everything. The claims that you cannot spread COVID if you're vaccinated were simply false. Vaccination can have a mitigating effect of course, but we were never going to be able to eliminate COVID with something like herd immunity.
The only virus completely eliminated by vaccines is small pox, and that was largely because of a number of ideal factors. The top two are probably that that no animals carried smallpox, only humans. And the second is that infection or vaccination provided lifetime immunity of effectively 100%. This dramatically reduced its potential for mutation and meant that getting rid of it in humans would get rid of it - period.
Coronaviruses, by contrast, are transmissible between humans and animals. This means even if you fully eliminated it in humans, it could, and probably would, come back. This is why flus are basically impossible to get rid of. If you believe CDC numbers then US flu deaths in 2020 were essentially 0, yet now it's back like nothing ever happened. Even the Spanish Flu is still with us as a variant of the common flu. But fortunately most viruses trend towards less lethal mutations over time. Probably natural selection in play - killing your host is not a great path to survival and reproduction.
As for long COVID. I'd rather defer that conversation for a few years. Research on exactly what it is and exactly what causes it is ongoing, and so debating it at this point is just going to be speculation.