I must be feeling argumentative today. "Ssssssomebody stop me!"
I believe the intent was to avoid many-to-most instances of "both sides roll" in the belief that roughly half as many people calculating their results would result in more streamlined game play. It can also work for streamlining stuff, if the DM knows the PCs' passive scores across the board and treats those as floors for determining, e.g., what characters know; you can think of it as being helpful for determining auto-success.
Players don't declare checks at all. . . In order to apply advantage or disadvantage to a passive check (e.g. disadvantage on passive Perception for traveling at a fast pace), the designers had to go with some kind of static modifier. . .
Passive checks seem to have an origin in 3e, a house rule which Chris Sims called "skill scores."
Rules Compendium said:
Every player adds 10 to each skill modifier to come up with what I call a "skill score" . . . Skill scores are useful for passive Listen and Spot checks too.
The problem arose when a PC tried to climb down a "narrow shaft" and failed a (unrequested) roll for it. Both player and DM agreed that no failure was necessary. The solution was that the character's skill score was above the DC of climbing, so the PC would have passed his passive check.
There's no need to use dis/advantage for the original passive check purpose, because it's a comparison of a PC score against a DC set by the DM. If there's advantage, the DM would just use a lower DC.
But I digress. There's no need for passive checks in 5e, because as iserith said, players don't declare checks at all, and DMs call for a check when there's "a chance of failure." Since a passive check also determines if there's a chance of failure, it's a redundant system. Yes, it can save time by suggesting that some die rolls aren't necessary, but it also takes time to compare one or more DCs to the passive score of each PC involved.
In the sleeping dragon example, as written, the OP wants to know if a dragon would notice sneaking PCs. That's a "contest," with the dragon's perception opposing the stealth of the PCs. But the dragon's asleep, which is obviously a disadvantage. No passive check needed.
But if OP considered that contest a bit too "active" for the dragon, one could just set the DC for waking the dragon (be careful, Daenerys) and let PCs roll against that. No passive check needed, no passive dis/advantage modifier needed.
Rule 0 is just the fact that you're allowed to make houserules.
So GMs don't actually run the game - the (book) rules do? Sounds like a lot of page-turning.