Ok I've already read through your entire post, twice, in order to give you the full respect you're due before jumping in and interrupting your flow.
This is clearly a point of contention as we're already not on the same page when it comes to what "property" means. I see now you're reading it as being a synonym for "possession". And, of course, it CAN mean that. OTOH, I've always read that context to mean something more along this definition:
Property is any item that a person or a business has legal title over. Property can be tangible items, such as houses, cars, or appliances, or it can refer to intangible items that carry the promise of future worth, such as stock and bond certificates.
Now, if we're not going to agree that SINless can indeed buy "possessions" that are not "property", as defined along the lines of just above, then we are indeed starting from points of little mutual agreement. You obviously haven't been presuming that a commlink can be a possession but not be property, whereas I have been. I can't speak for a 4e writer's intent, but I'm confident predicting that was the RAI but again I cannot say with authority there. So, we may end up agreeing to disagree, and if so from this point on our points of view will likely not reconcile any closer.
But let it be said that from here out, I'm operating under the definition that "property" is something that involves recognizing a title, or, more relevantly to the SR rules... licensure. SINless absolutely can buy drek from a vending machine and/or black markets. Furthermore, commlinks can absolutely go untied to any SINs, even for the SINners. It'd be analogous to a prepaid/no-contract cell phone in real life, and would almost assuredly be how any criminal would maintain their commlink service.
We are not at cross definations here... let me ask you this:
SINless guy walks into a store, and hands the clerk $2500 in certified Cred. The Clerk hands SINless Guy a Commlink....
How does the SINless guy take LEGAL possession of his property??
Short answer is: "He can't".
YES he has the plastic goodie in his dirty SINless hand... but he doesn't actually own it. (from a LEGAL stand point). YES he paid cash money from his own savings, the clerk acknowledged and affirmed the transaction by handing the SINless Guy the Commlink... But he doesn't LEGALLY own it. (Not in the world of Shadowrun!)
The reason that he doesn't
own it is.... Because he doesn't have a SIN! Full Stop. Everything you own is Registered to a SIN... Remember? I quoted it to you from 2 different sources (and while I don't have the 6e book, I doubt its changed).
As I have been saying, all your property, be that physical, digital, intangible, is tied back to a SIN. Heck, even the initials in your name, city of birth, date of Birth, GPA, credentials, schools attended are all Stored in your SIN, even your DNA... A SIN is the depository of your life.... The digital thumb print that you exist.
Without a SIN, there is very little a person can do in the SR universe that interacts with "normal" society...
Now SINless Guy can get around some of this by using a Fake SIN... and have the ownership of the Commlink transferred from the store to the fake SIN, sure. Now he technically "owns" the commlink... for as long as that fake SIN stays active. (And once it's burned, then everything registered to it is also burned.. And he "owns" nothing, even though he is in "possession" of the commlink...)
AND YES, Anything that is registered to a fake SIN that gets burned gets legally frozen (And probably flagged as "stolen"). Be that apartments, guns, clothing, cars, GRID access, Commlinks, and any other property registered to it. So get out those "dice" and start transferring ownerships
<Not that many tables go THAT far into the whole system and issues with SINs, but yes, RAW that is what happens. Most tables just hand wave all this away when a SIN gets burned... or should I say
IF a SIN gets burned... as this seems to happen very little in actual play.... >
And then there is the Matrix... which we have talked about and probably never going to see eye to eye on...
If the Matrix and SINs are not attached: How do you buy your GRID access? (they cost money remember? Except for the Public GRID... Eww! Loser). How do you buy your Music? Or your games? Or pay for your groceries? Or for that paywall to "Hot Troll chicks with Dwarf Girls on llamas"

? (Cause you GOT to see THAT! its a classic you know!) So clearly, there is
some overlap and cross connectivity between the Matrix and SINs... there has to be, or the Matrix is not a financial tool. (Because Bank accounts are stored in SINs remember?). How do you take ownership of your digital property if SINs and the Matrix Don't interact? And since they have to interact with your personna (because you're ON the Matrix!) to conduct the transaction....
See where I am going with this?
By the way, I don't know if you realize it or not but "magical matrix fairy dust" actually isn't actually that far off from being how it truly does work, in-universe. The corps actually don't understand how the matrix works... completely. More on that downpost...
***
Big Brother doesn't use "whatever technobabble the matrix uses with personas" in SIN verification tech because they don't understand how the Matrix works.
Wait... WHAT?!?!?!
"They" (being the Corps) build digital worlds in it, use it for data storage and transmission, conduct phone calls, digital meetings, build programs to run in, modify, edit, deny access to, encrypt and change it, transfer trillions of credits through it,
HOURLY. Use it to send thoughts, smells, images, sounds, and tactile impulses directly to and interacting directly WITH THEIR BRAINS (and yours too if you hot-sim VR).
But "They" don't know haw it works. Entirely...
Wow.... you would think, what with everything they can do, and DO do, they would know how it works...
I mean really

Now, I
don't expect there to be a giant blurb in the rulebook that goes into great detail about the minute details of
how it works.... but "
They don't know how it works, entirely"

Yea... ok.
This just doesn't work for some of us.