His success was predicated off of Alvarez's recruits and was on his way down when they left.
The degree to which this is true is debatable, definitely true for 2006 but the other years? I don't know.
2006: 12-1, All Barry's players
2007: 9-4, Still Barry's players
2008: 7-6, Mostly Barry's players
2009: 10-3, Mostly Bret's players
2010: 11-2, Mostly Bret's players
2011: 11-3, All Bret's players
2012: 8-5, All Bret's players
Bret surely inherited the offense's system from Alvarez and kept it with little change and he definitely benefited from Alvarez being the AD as Barry has great coaching connections and is a master at accessing coaching talent but to say BB's success was predicated or based solely on Alvarez's recruits isn't entirely true. Two of those teams of Alvarez kids didn't do very well.
So what you should say is that BB's success was based on Barry's tradition and oversight generally with one huge caveat.
The key to Wisconsin having a great year rather than a decent-to-good year, and the difference between Alvarez and Bielema, is quarterbacking. Those big O-Lines, prolific RB's and staunch defenses are always there more or less but a good QB puts them over the top. Bret was only able to get one good QB on his own that he wanted and that was RW off the bounce from NC State for a year, other than that he struggled. Barry had Bollinger, Sorgi and Stocco, hell even Tolzien was an Alvarez recruit.
I see no upside to having Alvarez as a coach.
What?
I agree with you. Although one of those Rose Bowls were a direct result of OSU and PSU being under sanctions.
Direct result? No. That Rose Bowl was a direct result of Wisconsin winning the Big Ten Championship game 70-33. Wisconsin's presence in that game was a direct result of Ohio State's rampant cheating and Penn State committing one of the worst cover-ups of child abuse in American History...that and their winning the games that counted.
Petrino did a LOT with what little talent he had.
Wisconsin under BB did this to a greater extent than Petrino did. I don't know if that was BB or his assistant coaches but it shouldn't be an excuse that Petrino was just some god at developing talent and it can't be done.
Bielema had a big talent deficiency from Wisconsin to Arky.
Arkansas had better recruiting classes than Wisconsin every year from 2002-2012, often significantly better (Arkansas avg 27, UW 48) so even with player attrition he's still walking into, from a recruiting perspective, a more talented team at Arkansas...though compared to the rest of the conference much worse.
I agree that Petrino's departure hurt Arkansas' roster and they are surrounded by schools that can sell themselves better (OU,OSU, UT, TAMU, LSU, UM even OM at this point) but that alone doesn't explain Bielema's atrocious season...I think that's due to his offensive approach. AR Husker Fan is right in that Bret's (Alvarez's) system doesn't work in the SEC if you don't have players on par with your opponent. To expand on his point look at the SEC and their offenses:
Pro
Alabama 11-2
LSU 10-3
Vandy 9-4
UGA 8-5
UF 4-8
Arkansas 3-9
Multiple (mostly Pro)
South Carolina 11-2
Ole Miss 8-5
UT 5-7
Spread
Missouri 12-2
TAMU 9-4
MSU 7-6
HUNH
Auburn 12-2
Air Raid
UK 2-10
Take from that what you will but most teams (9/14) are running a Pro, or mostly Pro-style, type offense. It will be tough for Bret to win games with his current style and recruiting as poorly as he has been lately.