Not true. The flu vaccine is pretty poor against infection.

Some vaccines do prevent infection. They are typically against viruses that mutate *extremely* slowly such as measles.

Many do not. It's extremely hard to achieve sterilizing immunity against coronaviruses or the influenza group or...quite a few others. What we can achieve is resistance, and resistance is enough to end the pandemic, even if it does leave us with one more respiratory annoyance.