No, most definitely not. Vaccinated individuals are hospitalized less, are intubated less, die less, quite simply, they use up our healthcare system less. Lockdowns are in essence a way of preserving that limited resource (hospital beds).
Where I live, in Alberta, we had a really bad 4th wave in September/October. Prior to that we were at ~75% vaccination rate in those over 18. The wave almost killed our healthcare system and it was primarily unvaccinated individuals using the beds. Had we been at 100% vaccination rate, we most definitely wouldn't have been in the same position.