I agree that "breakthrough" is a lousy term.
But vaccine is just the word for something that activates the immune system to fight a specific thing. If we rename the covid vaccines to something other than vaccines, anti vax people will use this as some other lame reason to avoid shots. Not all vaccines are perfect--our flu vaccines are much less effective than our covid vaccines.
One thing that makes all vaccines more robust in the general population is if everyone gets them. Every time one of us gets jabbed, it reduces the chances of covid to find another home, spread, mutate...
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