Reaver, Seras is having issues not with SINs, but with people not having SINs. Because he's from Germany, and they tend to track things ... rather closely.
Seras, to help you out in your disconnect, remember that society has fragmented more than a little. Consider a home birth, or just one that happens 'en route' or something; in order for the child to be documented (which the hospital staff would normally take care of), the parents have to specifically go about and get it done, whether that's taking the baby into the hospital as soon as they can (and giving information then), or else going about to ... I honestly have no idea where. Where-ever. But if someone a) doesn't know, or b) is too lazy, then c) you get someone whose information isn't entered into the databases. If the parents are criminals -- with all the necessary societal tracking they want to avoid -- then getting their kid linked in officially becomes an issue, because it gets the hooks into them as well. So they avoid it. And if society for some reason doesn't want 'their kind' in, from orks and trolls in this area to elves in that one and immigrants all over, then they can actually be denied a SIN, and become 'provisional' citizens -- SINless.
Your main disconnect doesn't seem to be the SINless, though -- it seems, really, to stem from how easy it is (or seems to be) to acquire a false SIN.
Truth is, a good fake SIN (4+) is not actually all that 'easily' accessible; at Rating x 3, that means they're as tough to get a hold of as a cyberdeck of equivalent rating. While most GMs waive the test, realistically you've above a 50% chance of getting your hands on a L6 ID only once your Negotiation + Charisma pool goes 20+ -- because it's an Opposed test against the Availability Rating (18), and you need at least 1 net hit to get it. Yeah, maybe you're checking once a week because you're paying only 10-15k (which is, in my opinion, far under the amount you SHOULD be paying for a high-quality piece of gear that'll beat a L6 SIN scanner 15 times out of 16, and only need inspection one in six times -- and how often are you likely to run into a L6 SIN scanner?) but that still means that your negotiator with his 15-dice pool is still going to take an average of 3-4 weeks before he can manage to get his hands on it. Even that R4 SIN isn't likely to be readily accessible; that's a 58% chance of availability (i.e. 1 net hit) for said 15-dice pool negotiator to acquire.
Ideally, you or your GM simply roll a few times, to figure out how long it's going to take, and then the guy who's getting you the SIN takes what information is needed -- usually biometric stuff, but often a quick rundown of your own background and shopping habits -- and gets it to the people who do this for a living. They look through their list of SINs for a decent match, and depending on what you need, they might either update the SIN's biometrics to match your own, or else they return a 'close enough' SIN. Those might be stolen/replicated, they might use dead people, or they may use one of the many, many SINs they're constantly planting and updating.
A good fake is critical, and though getting one is usually glossed over, it is not a 'gimmie' -- just try to get one using the rules, i.e. in an emergency ...
All this having been said, creating a false front for 'legal' shadowrunning is a time-honored classic. Almost every group I've been in has done it: everyone acquires a L6 false SIN, pools their resources, and with the fake SINs creates a corporation that is, essentially, also fake. Oh, it may have real employees and all; my favorite is the company with couriers on the one hand (whether that's street runners or high-speed vehicles) and security/bodyguards on the other (for all your mystic and muscular leg-breakers). You might branch off into high-threat medical rescue, charging big bucks indeed for going into those places even DocWagon won't pull you out of ...