It's actually quite the opposite.
As mentioned, Miller gets extra attention, gets chipped, etc. Thus, leaving Chubb often one-on-one with a tackle or a tight end. It's a benefit, as you say.
For a corner, playing opposite of a shutdown guy could get you exposed because they aren't going to throw at Fuller. Example, right now, no one even throws at Ward. Instead, they pick apart the likes of TJ Carrie. If he was opposite of a shutdown guy, he would get thrown at much more frequently. Thus, not necessarily a benefit for the player himself.