If there is some protection from spread, it's garbage protection.
Since you seem a bit confused, here's what we know and don't know AFAIK:
1) Vaccination reduces your chance of getting infected. The efficacy depends on the strain of covid and the vaccine, but I have yet to see any science indicating that any vaccine has no efficacy against any strain.
2) Vaccination may or may not reduce your infectiousness while you are sick
3) Vaccination does reduce the average duration of infection, which means that while you may be as infectious as a non-vaccinated person while you are sick, you have a shorter window in which to spread that sickness.
So despite the unclearness of #2, we are very clearly able to say that vaccines reduce transmission because of #1 and #3.
I believe that the emphasis on the reduction of the chance of serious illness & hospitalization is because people believe that all the people who would be motivated to get vaccinated out of altruism are already vaccinated.