We are probably in the middle of the road on this one. The septic system on an RV is not the same as the SticknBrick, and anyone who says it is either has never actually spent much time RVing or didn't understand the OP question, or maybe both.
We don't use a screen in the bottom of our sinks, but that isn't a bad idea. We are careful and try not to let grease and large food pieces get down the drains.
I go through my regular flush cycle when dropping my black tank, followed by the two gray tanks. I will also close the gate valves to the black and one gray, leave one gray open, and use my Flush King to flush that gray out several times. Then I will reverse the two gate valves and flush the other gray tank out several times. On occasion we will use a gray tank additive to closed gray tanks also.
I just think you have to use some caution on what goes into the gray tanks and no matter what they should be flushed out once and awhile.