Bob, the area along the first mile of the Abrams Creek Trail is not too bad for someone to stop and read while the other is fishing, but the rhododendron canopy is quite thick in many spots and I think there are better options. And yes, the stream bed is pretty slick (more so than any other stream in the park).
If it were me, I'd go somewhere up on the Little River Trail above Elkmont. The Little River Trail is fairly flat and has lots of good areas to stop while someone else is fishing. Plus there are a number of side trails that you can also hike, when you're not fishing.
I would normally also recommend the Middle Prong Trail, but because you are coming on June 4th, you would be there during the Lynn Camp Fish-out and it might be crowded.
"Even a fish wouldn't get into trouble if he kept his mouth shut."