“The claim that the mRNA doesn’t go away and continues to produce proteins....”
mRNA molecules are by nature fragile and breakdown easily. That’s why figuring out how to protect them long enough for them to do something useful is A Big Deal.
That process incidentally is the gist of Robert Malone’s 1989 paper, which he likes to say makes him “the inventor of mRNA vaccines”. No one else in his field seems to think it makes him that, but it was an important step.
The vaccines use nucleoside analogs and have changed the mRNA sequence at to avoid the breakdown enzymes.
The vaccine mRNA does not breakdown naturally, and Pfizer and Moderna consider that a benefit.