There is nothing preventing a runner with the SINner negative quality from earning and associating legitimate licenses with their legitimate SIN.
Core Rulebook, page 419
Licenses and permits can be obtained through legal channels, as long as you’re a solid citizen with a legal SIN—of course, that goes for shadowrunners with fake SINs acquiring fake licenses, too. A fake license is connected to a fake SIN, and if one of them is exposed, the other becomes worthless (see Fake SINs, p. 367).
Core Rulebook, page 367
A basic SIN allows a person to function within the most basic parameters of the law. If a person wants to operate outside these very confining strictures, they need special permission in the form of a license. What actions or items require a license vary widely with the laws of the country, but there are some common similarities shown in the Common Licenses Table below. Acquiring a license (legally) is, again, dependent on the laws of the country. For the most part, acquiring a license involves making an application, paying some sort of fee, a SIN check, and possibly passing one or more certification, testing, or training programs. The exact details of acquiring a license are left up to the gamemaster, but government bureaucracies are notoriously slow and frustrating.
The biggest problem, in addition to those you point out yourself, is that not every act can be made legal with licenses. Only Restricted items can be made legal to possess with a license, and technically according to the Core Rulebook you need a separate license for each individual type of item (which means you need a separate license for your Katana and your Shock Gloves to stay completely within the law). The first time you get caught with that Forbidden Cavalier Arms Crockett EBR or FNP93 Praetor then both your legit SIN and all the licenses associated with it are burned. So it's basically a complete waste of time and effort. But sure, if you want to make the game more difficult, you could do it. If it were me I would restrict myself to Restricted gear despite all the best weapons and armor being Forbidden to make the choice slightly less dangerous, and I would use exclusively Stick-N-Shock ammo for living targets to avoid committing murder whenever possible since sooner or later your illicit activity will catch up to you.
Also, while fake licenses merely need to be purchased, your GM may make you jump through additional hoops to earn legitimate licenses. I'm reminded of the episode of Star Trek: Voyager (S07E22) where Tom Paris is accused by the locals of reckless piloting and is forced to take what amounts to a "safe drivers" course in order to regain his permission to fly within their space.