For the same reason we have tax discs. They replace the whole plate every year, we replace a small piece of paper.
No, we do a similar thing as you, although implementation varies from state to state. In some states, there's a sticker that gets put on the plate, in others, it's just the actual registration card that gets changed, but AFAIK, there's no place in the U.S. where the actual plate changes year to year. As another poster pointed out, the main difference really is that here, plates are usually assigned to a person (I say usually because someone else mentioned California being different), meaning when you buy the car (new or used), you get plates for it at the DMV and keep them, usually until you get rid of the car. You can also (again, may not be true in *every* state) keep those plates if you want and put them on your new car rather than turning them in and getting new ones (probably more popular with vanity plates than normal ones, but you can still do it either way). This compared to other countries where the plate is permanently attached to the vehicle and is reassigned to the vehicle's new owner when it's sold.
You'd still really have to fail the attitude test in a big way to be arrested for an out-of-date tax disc.
I'd say the same thing is normally true in the U.S.. I've known plenty of people who were caught driving with an expired registration, and unless there was something entirely else going on (outstanding warrant, open container, drugs) I've never heard of any of them getting more than a fine, and perhaps being told the car has to be towed.