Since Omicron, completely true.
Not true at all with Alpha and to a lesser extent Delta.
There was about a 2-3 month period in November-January where it became really clear that the vaccine was not having any significant impact on infection where people were still acting like it did but I honestly haven't seen anyone claiming it did in recent months. A new vaccine is needed for those wanting it. By now, case fatality rates are so low and almost everyone has some sort of immune response we are no longer dealing with a novel virus.