Asymptomatic really just means you don't have symptoms at the moment, not that you never will. I've always assumed when they said asymptomatic infection spreads the virus, they really meant pre-symptomatic.
Someone who 12 hours later has a fever and body aches very well could have been spreading that throughout the day.
That's different from someone who has very low levels of the virus and their body fights those low levels off quickly (whether because of the vaccine or their natural body defenses)
No they really meant truly asymptomatic spread, those that never developed symptoms it was all over the news last year and was the main concept they used to convince people of the need for masks.