The term "system quarterback" is a pejorative...it implies that the player has deficiencies, mental or physical, that are successfully mitigated by the system in which he plays.
I agree with slaughter25 that Alex Smith it the best example in the NFL today. Harbaugh implemented a system that plays to his strengths and away from his weaknesses, particularly limiting his downfield throws and emphasizing safe passes and throwaways, thereby limiting his interceptions. Smith has the physical tools but is not the smartest knife in the drawer, football-IQ-wise. Harbaugh got his QB rating up to 91 last season from a previous best 82 on the strength of limiting Smith to 5 picks. You can win that way...except for two muffed punt returns by a last minute substitution and those guys go to the SB.
Another single-season example, admittedly controversial, would be Brett Favre in his best statistical season...2009 with the Vikes. Childress prohibited him from calling audibles (a grand total of 3 for the whole season, by one account). As a result his interception % was about 1/2 of his previous best. For this brief time, the Childress system dampened Favre's "wild hair" tendencies.
However, I think what you're looking for are the more commonly cited career system QBs who had a lot of success, which would include two guys with a lot of similarities...Joe Montana and Tom Brady. Coming out of college, these guys got poor scouting grades for having bad bodies, mediocre arms and poor mobility, and were drafted accordingly. You might as well toss Bart Starr into this category. Brady got tagged "system" when Cassel put up a nice season in his absence. Montana got benched briefly in his early prime for failing to implement "the system", and the franchise did not miss a step with Young after Montana's departure. Montana, Brady and Starr also happen to have played for legendary, innovative coaches. The "system QB" tag rationale with these guys is that they MUST be system QBs because they don't have outstanding physical measurables. To those critics we must respond that they just might be measuring the wrong things, or more likely overlooking the things that are not measurable, such as a deep understanding of the game.
Looking at Rodgers, the physical tools can't be an issue. Whatever mechanical deficiencies he had coming out of college were long gone by the time of his first start. His mobility is not an issue; he throws on the run better than anybody. You would not think football IQ is an issue. While he doesn't call his own plays as Peyton Manning has much of the time, he's run a pretty spiffy hurry-up offense. With Saturday out there, we might see more of that. Besides, you don't put up those kinds of numbers with so few interceptions without a superior grasp of the game. That pretty much leaves the Matt Flynn factor.
If Flynn's brief performance is all that the doubters can point to, we can only respond that Flynn's record is not statistically significant. Or maybe he happens to be a pretty good QB in his own right...he looks like he could have a Matt Hasselbeck-type career if Seattle comes to their senses and hand him the ball.
The question you need to ask with Rodgers is, "what deficiencies are being masked by the system?" Personally, I have a hard time finding any at all. "He holds the ball too long" is a frequently noted criticism. That doesn't make a lot of sense when you look at his passer rating when he breaks the pocket. The guy performs in the system and when he breaks out of it.
In the final analysis we have to ask, "so what?". There's no way of knowing what these teams might have done with somebody else at the helm. What counts is the numbers you put up, particularly the wins column.