iserith
Magic Wordsmith
From the PHB:
"A passive check is a special kind of ability check that doesn’t involve any die rolls. Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster."
When I first read your post, I thought you were wrong about the "always-on radar", but after rereading the PHB, I'm somewhat inclined to agree...
For the case of hidden monster, it does seem the interpretation of "always-on radar" is correct. However, with regards to finding a secret door, it could be read that a player will tell the DM they are searching for secret doors down a corridor, so instead of rolling every 5 feet, the Passive Perception is used. In this case, the term "passive" becomes a misnomer and confuses how it's supposed to be used.
You could also argue from the above quote from PHB that only when a task can be done repeatedly - which assumes failure is not "dangerous" - can a passive check be used. In other words, I don't think it's unreasonable to rule that traps CAN'T be found with Passive Perception if you don't want them to be, or at least not without special circumstances.
I think the best way to look at it in terms of process is as follows:
1. DM describes the environment, including clues that telegraph hidden things.
2. Player describes what he or she wants to do.
3. DM decides if the outcome is certain or uncertain. If certain, go to 4. If uncertain, call for an ability check if it's a one-off task or a passive check if it's a task being performed repeatedly.
4. DM narrates the result of the adventurer's action. Go to 1.
This is consistent with all of the D&D 5e rules in my view and works in all situations. It's also dead simple.
Thus, if a player describes wanting to keep an eye out for hidden monsters and traps whilst traversing the dungeon, the DM is likely to consider that to have an uncertain outcome and, since it's an ongoing effort, apply passive Perception whenever necessary to determine what happens. The character also has to be in the fictional position to notice the monsters and traps. In many cases this may mean in the front rank of the marching order and the character must be moving at a Slow pace (unless the PC has the Dungeon Delver feat) which can be risky. In my campaigns, for example, moving at a Slow pace means your chance of running afoul of a wandering monster goes up.
Performing any other task that is at least as distracting as Navigating, Drawing a Map, Foraging, Tracking, or the like means you are NOT keeping watch for hidden dangers. You're automatically surprised by lurking monsters or you blunder into the trap as appropriate. In my experience as long as other tasks are at least as beneficial as keeping watch for hidden dangers, you will see players choose these other options depending on their party composition and priorities. This further diminishes the "problem" of passive Perception being "way too important."