Yep. That is the point and the main idea behind vaccination. And, in most cases the production of neutralizing antibodies against the vaccine target is enough to protect yourself from infection and to prevent you from being able to spread the disease as well. However, that is apparently not the case with SARS-CoV-2 and Covid19 disease. While neutralizing antibody titers in individuals who have been vaccinated were high and at levels that would be expected to prevent carriage or transmission of infective virus that has not proven to be the case. Instead it appears that for Covid19 (at least for some or possibly most people) the production of neutralizing antibodies alone is not enough to provide complete protection for you and everyone that comes into contact with you. Why that is the case is an open question though there are some hypotheses and it gets very complicated very quickly. Without a deep understanding of immunology it is difficult for me to communicate what they are, but suffice to say it has to be the right "kind" of antibody (and I don't just mean the right class) and it may be the case (emphasis on may) that whole spike protein has a fatal flaw as a vaccine target. Finally, and I shudder to think of the consequences if this were to be accurate but there is precedent for the production of the “wrong” type of antibody resulting in a worsening of disease in a process known as antibody dependent enhancement or ADE. This is typically only an issue if vaccinated individuals are then re-exposed to the agent against which they have been vaccinated. There have been vaccines in the past in which this phenomenon has occurred with horrific consequences for those effected. There is zero evidence of this at the moment but it is a hypothetical possibility that cannot be ruled out or shrugged off as impossible.
Incidentally one of the ways the vaccine manufacturer's have been making "effectiveness" claims is by touting the correlation of their vaccine with the production of neutralizing antibodies in vaccinated individuals. I do not if that practice was part of the most recent trials because those details have still not been made public. However, if they are doing this than their claims are wildly off base and will be misleading in the extreme.