Of course they do. Have you ever stood in a tower and seen how much of the air traffic system relies on the pilots? Watching the game from both sides, I can say with some certainty that the perception on both sides is that the other side "has our back". In this case, as an approved procedure, ATC will not think twice assigning it to anyone. By accepting the clearance, the pilots are agreeing they can fly it. If they can't or if there's a problem with the procedure, it's not ATCs area of responsibility.