To me this discussion is making a bit of a mountain out of a molehill. I get that DM's have a lot on their plate to remember. Unfortunately that's part of being a DM. I've DM'd pretty much exclusively since around 2005, before that I was almost exclusively a player, so I've seen both sides.
There really aren't that many passive abilities that the DM needs to remember and roll for. Off the top of my head they would be elves detecting secret doors, dwarves and stonecunning, passive spot and listen checks, possibly passive search checks. I'm sure there are others that could come up. For example, if an NPC is lying to the party, they should probably automatically get a Sense Motive check to see if they pick up that the NPC isn't being completely truthful with them.
So as a DM, make a note of these passive checks and keep it somewhere that you can see it when you DM. That should help you remember until you get to the point where you've done it enough times to just remember automatically.
Of course, no one is perfect. I forget stuff in my game all the time as DM. However, one thing I try and keep in mind when DM'ing is to not sweat the small stuff. So if you forgetting something didn't get a PC killed or cause the PC's to miss out on an important clue or piece of the plot, then it's probably not big enough to worry about. As long as everyone is having fun it's all good!