No qualms with that. You can't be engaged with another activity and be attentive and alert with your surroundings.The activities the rules call out as obviating a passive Perception check for keeping watch are navigating, tracking, foraging, and map-making. Any similarly distracting activity could do the same. They key thing here is that there is a potential trade-off. You can continually keep watch and gain the benefit of your passive Perception or you can do some other activity and potentially not gain that benefit - choose.
This is where I disagree slightly. Just like someone using passive perception shouldn't have to specify that they are looking for traps or stealthed creatures, someone using passive investigation shouldn't have to say "I'm checking for illusions." That to me says more of an active check with dice rolling. Or if you want to take it a step further, a barbarian who really suspects illusion magic taps everything with his maul.I read it again and I don't infer that from the feat. Sure, if a monster is taking steps to remain vigilant for the telltale signs of illusions, then the DM could reasonably rule that passive Investigation applies, if the outcome the monster's effort is uncertain. But this level of scrutiny (which probably includes more than just gawking) should come with a trade-off in my view, as above.
In other words, the fact that it's passive means you're scanning anything and everything for abnormalities. A higher score means you catch more things.
Treating passive Investigation as a form of "always-on radar" strikes me as an attempt to justify making illusions less effective.
Well yeah, that is what it is. It seems to me that a well executed illusion has no counter if you don't use passive investigation. Let me use an example:
An enemy party is holding a noble hostage in his estate. The party consists of two rogues, a wizard, a barbarian, and the leader. The leader, a fighter, says to the party, "Stay here and keep an eye on our hostage. I'm going to check on our sentries posted on the upper levels. Don't do anything until I tell you." The players have a party with an illusionist and gets word from the scout. The illusionist makes a minor illusion of the leaders voice saying "Everybody get up here, there's trouble."
Now there's a few different ways the DM can go about this:
1. The enemy party all rushes to action. The PCs rush to get the hostage out uncontested.
2. The party all makes investigation checks for the source of the sound, checked against the spell DC. The player could argue that the party had "No logical reason" to investigate the sound and they do have somewhat of a point since the leader did just say wait for my next order. Not to mention the enemies would not want to "waste action" investigating when they think they are immediately needed to deal with a threat.
I wouldn't have a problem with the "meta knowledge" solution of NPCs rolling against every player illusion. In fact it seems many DMs do this for stealth and disguises. This does have the effect of slowing the game down however.
3. The enemy wizard is on level with the PC and automatically detects the illusion due to the passive investigation. This creates a new dynamic of having to scout for wizards and high intelligence creatures when attempting illusion trickery. And that perfectly makes sense, those creatures are practiced in magic and should be better at detecting illusions even when they aren't expecting them.