How do I install a single-pole switch w/ pilot light?
So, after some further fooling, I rewired it again as I did the first time (with a pigtail going from the neutrals to the silver terminal), the hot line in going to the top right black terminal, and the load going to the bottom right brass terminal. I turned on the breaker and again, the pilot light is on 100% of the time, whether the switch is in the on or off position. 
So, I then swapped the hot line in, and the hot load out, and turned back on the breaker, and  it now works properly. So, it looks like the directions are SORT OF correct, but Leviton put the line in where the load out should be, and vice versa. Basically what they say should be connected to the black screw should be on brass, and brass should be on black (still a DIYer, I never know the proper names for these things). I hope this helps someone else who buys this same switch.