Consumer fraud protection in the US means you're not liable if they copy down your details. And the companies seemingly would rather do it this way, it saves money in the end, even though any fraud that happens raises their clearing fees. Remember, there is nothing stopping US restaurants from bringing a portable transactor to your table. Those things read swipe cards and PIN cards just fine. So if they aren't doing it by choice, there could be a good reason.
It does reduce waiter back-and-forths, but is that really the limiting issue? The waiter bringing the reader and waiting while you use it increases waiter time spent which costs money.
If you want to go fast, ask your waiter to do the job fast. Otherwise, the restaurant can save money by having a pile of those little trays/folders and waiters picking up and running 3 at a time.