Yeah, the Psychic Tradition is basically just kind of there to show that even if you have no clue what you're doing and are basically 100% wrong, if you're Awakened, you can do magic. They are an extreme example of "the bones of what we believe".
Now, people who would be called "psychics" but aren't magicians of the Psychic Tradition are usually just the "Aware". People who are technically awakened, but the most major thing they're capable of doing, magically, is Astral Perception. Most of them have small magical "quirks" more than anything, some don't even realize they're seeing the astral, since they don't have enough talent to be scouted and trained, while others end up doing the whole "palm-reading fortune teller" thing.
Deciding that genuine psychic power, like purely mental telepathy and stuff that isn't magic, would be far outside the realms of Shadowrun. I'm sure a lot of people called "Bullshit!" when technomancers became more blatantly mystical (instead of the just slightly mystical Otaku), and they were just a progression of an existing feature of the game. Trying to fit "suddenly, there are non-magical psychics" into the world would be a very big task that, as you said, would probably fall apart after only a bit of play... It probably isn't worth the huge effort it'd take.
However, if you want to look for unique and subtle threats, there's definitely stuff there for you to use. I'd look into the various kinds of spirits that exist, or if you wanna avoid something magical, maybe look into AI and learn about what the whole Renraku Arcology thing was like. I'm sure others on the forums have better suggestions too.