Exactly. That's why we have courts to determine the facts of the case and the suspect's intentions, as well as whether or not they did it innocently (i.e. receiving a counterfeit bill from someone else and failing to detect that it was fake) or with intent to commit forgery.
Executing someone in the street, in broad daylight, means that the suspect's innocence or guilt cannot be determined. The whole point of arresting someone is that they have to be, uhh, alive. Last I checked, you can't arrest a corpse.
I mean, I suppose they could've done the whole Weekend at Bernie's thing and tried standing the body up before a court. Not sure how that would've gone over with the protesters, but hey, they could've given it the ol' college try.
My point wasn't that arresting him was inappropriate. My point was that it's ridiculous for a man to die over a fake twenty when the Federal Reserve is currently counterfeiting trillions of dollars to fight the effects of COVID-19 on our financial system, and last I checked, no one has knelt on their necks for it.