As to whether the claim is accurate... that's outside the scope of this comment.

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.

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.

If there is a part of my comment you found snide, please let me know as that was not my intent.

I have never (that I recall) seen an official source saying what you claim. (Mind you, I'm not saying that you're wrong in your assessment of the effectiveness of the vaccine. I'm not even saying that you're wrong in your read of what officials are claiming - just that I have never seen it.)

Can you point me to someone official making that claim? And, what country are you in?

So.. It's not perfect at stopping transmission but you can see how it would definitely reduce the R0 enough to prevent outbreaks in the non-vaccinate pops.