Flu vaccines have been completely unable to eradicate the influenza virus in the way that other vaccines have. Furthermore, they have most certainly NOT been able to deprive the virus of hosts, with that whole 50-70% effective thing every season.
Your problem is that you probably just read your sons’ “Intro to Bio and other fun things too” textbook and you think you have it all figured out. Unfortunately, to actually understand virology, you have to go past learning what a capsid is and which ones are double or single stranded.
thanks for playing.