I almost feel that my preferred solution would be to ditch perception altogether!
If I were to keep perception, probably give perception a much higher DC (more than the difference between trained and untrained)
Personal opinion, of course, but I do think there's still a place for a catch-all "notice something" skill, it's just that it would no longer be the be-all-end-all skill that it is now. Instead it would be replaced by a generalised skill that provides generalised knowledge.
The question then is, because the skill is now weakened significantly, do you then still weaken it by offering different DC's?
I'm starting to lean towards yes. My reasoning being that once you alert the group to SOMETHING, they'll obsess over it until it's figured out, using every other skill under the planet to do so

Which somewhat defeats the purpose of specific vs. general.
So if you both increase the DC (I don't think the DC should increase by more than 5 as that puts it beyond the realm of reasonable achievement) and make Perception generalised, whilst being somewhat lenient in allowing other applicable skills to the situation give specific knowledge, the player can then choose between a skill that gives them a small chance at noticing ANYTHING, or a skill that gives them a very good chance at noticing very particular things.
Then you'd have the situation of Bob the Perceptive saying to his thief pal, "Hey, what's that thing?" Or Bob's thief pal, Tom the Trapspotter saying to Bob, "Don't step there, that's a trap trigger mechanism."