"It's my cousin's, he let me borrow it."
I think the main effect is that the device will immediately show up as stolen if it has been reported as such. But what if the original owner is dead or a criminal? Or both? It's not going to be reported, so it could be borrowed.
Of course if the police is suspicious and checks, and finds that the true owner is dead, they're going to have some hard questions for you. Avoiding that is where social skills come in, but at some point you're going to meet a cop who's not going to fall for it and check anyway.
So I think theft will work great for the short term, but won't be sustainable for long. (Please FBI, don't read me saying this.)
Yes Matrix forensics is going to be able to dig down and give the game up. But most of a Runner's gear is going to have traveled through some shady channels already. Intense Matrix scrutiny of an item's history isn't going to go well for a lot of things in a Runner's kit.
That's another issue: how much of your gear will stand up to that kind of scrutiny? Is it registered to one of your fake SINs? Is it the one you're currently broadcasting or one of the others? The police might think you've stolen it from yourself. Stumbling into an overly diligent cop at the wrong time could end up costing you two fake SINs, and not just one.
Disabling a device's wireless so it's not broadcasting any ownership info will work very well for easily concealable items. Not so much for a car, though.
May players err on the safe side and don't loot anything. Well, nearly anything; they just killed a serial killer who happens to have a slightly more upgraded version of the same rifle that the street sam had shot from his hands by an APDS round in the same fight, so they're going to see if they can combine both of them into a single functioning rifle registered to the right SIN again. Also, the killer had expensive ammo which would be a waste to just leave lying there.