> Vaccination is neither prevents infection nor spreading.

Vaccines frequently prevent infection and spreading. (And yes, they do this by way of priming the immune system.) For example, we eradicated Small Pox with vaccines: https://pubmed.ncbi.nlm.nih.gov/788150/

My understanding is that vaccines do reduce the chance of Covid spreading. But with the Omicron variant, they help a lot with preventing severe illness and only a little bit with helping reduce infectiousness. And Omicron is ridiculously infectious, so it's going to whip right through well-vaccinated areas just like everywhere else.

But in general, reducing the chance that one person infects another is one of the primary purposes of vaccines.