Replacements of HOF QBs (that retired as starters and did not move to other teams as backups) have met with lukewarm results. They may have had one or two good seasons, but none of these guys came close to their predecessors career wise . .
Jay Fiedler (Dan Marino) undrafted
Brian Griese (John Elway) 3rd round pick
Tommy Kramer (Fran Tarkenton) 24,777 career yards, 159 TDs
Mark Malone (Dan Fouts) Was not considered a good QB before taking over, and was in his 8th and final season as starter when taking over for Fouts
Steve Bono (Joe Montana) 6th round pick
Todd Collins (Jim Kelly) 2nd round pick
Jeff Garcia (Steve Young) undrafted, still went on to have some excellent years, 4 time Pro Bowler, 22,825 career yards, 149 TDs
Quincy Carter (Troy Aikman) 2nd round pick with a drug problem
Cliff Stoudt (Terry Bradshaw) 5th round pick
David Woodley (Bob Griese) 8th round pick
Scott Hunter (Bart Starr) 6th round pick
Danny White (Roger Staubach) 21,959 career yards, 155 TDs
Not sure what your point is. If it's that a HOFer isn't likely to be followed by another HOFer, that's not exactly a surprise or a revelation. You picked 12 QBs. Choose any random set of 12 starting QBs in the history of the NFL and you aren't likely to find a HOFer in that group, or in the group of QBs following them either. But even in your sample, 25% (Garcia, Kramer, and Danny White) had nice careers. That's not bad is it? Most of the others who followed the HOFers, based on their draft slot or the point in their careers when they took over, wouldn't be expected to shine.To me, the sample above has no relevance to Aaron Rodgers at all. It looks more like a conclusion already drawn about Rodgers, that he will fail, then a sample found that would support it. I don't think that conclusion is at all accurate.
Rodgers was a 1st round pick, unlike any of the replacements listed above other than Kramer (who did fine for a career) and Malone (who came anew to the HOFer's team after an undistinguished career already). Rodgers has 3 years of learning in this system under his belt, and is in a solid young offense. The closest player that comes to my mind was Steve Young, who also was a 1st round pick, sat behind Montana a few years, and eventually became a HOFer himself after taking over. I have to wonder why you qualified your sample to exclude HOFers moved on to another team (like Montana) rather than retired. What difference does that make in how the successor would do? Was this condition included purposefully to exclude Montana and Young because you knew it would keep Young out of the conversation?