mRNA vaccines are highly localized: you get a sore arm because most of it only gets taken up by muscle cells around the injection site, which spend some time producing the antigen and triggering a primary immune response (the inflammation aka the sore arm).
I can remember getting a sniffle at night and waking up fine the next morning a few times.
I think I had two doses of covid mRNA vaccine.
I have actually forgot what it is like to be sick. It almost feels like the covid vaccine gave me some kind of super immunity. I never get the flu shot either. I have not had the flu in 5 years for sure.
As for being localized it's true however after vaccine dose S proteins have been detected also in remote locations in the body because you can't make something 100% localized.
If you had an infusion that doesn't trigger immune system you could just increase the dose significantly, put it in the blood and most likely it would have reached all cells that blood reaches.
But I'm speculating a bit here.