Well, some things can't be rushed, the grass tops may die sooner, but the roots take longer to die, especially if there is Bermuda grass in the mix. You should be able to check and see when the right time is for you situation.
I don't have problems with fungus in the mulch, it is rather dry around here, I reapply the mulch once a year.
Yes black plastic will kill the grass but it does nothing for soil improvement, and it blocks the water. I like the fact that paper and cardboard will decompose and add organic matter to the soil, the worms love it.
I guess we all have to do what seems best for ourselves, I have never been bothered with the grass coming back and it has been three years now.
