Though they did get lucky that the Colts rested Peyton Manning when they did, they dominated the Bengals when they had all their starters in week 17 and beat them handily in the playoffs. That was also by far the biggest break they had this season, and the only significant one. For example, in week 8 they held the entire Dolphins offense to 104 yards while gaining over 300 themselves, but lost because Ted Ginn returned two kicks for touchdowns. I'd hesitate to call it luck, but it is a break and a fluke.
Mark Sanchez may not be a great QB, but he has some great tools and when he was able to get into a rhythm he was very accurate and poised, and he did play well in the playoffs. Obviously he had his problems, like throwing 20 interceptions, but 11 of those 20 picks came in just three games, so that's 9 picks in 12 starts. If you expect much more from a rookie QB that started less than 15 games in college you have a LOT to learn about the NFL. I hope I don't come off as condescending, but if you think that guys like Matt Ryan and Joe Flacco come along all the time you're mistaken (and both of them were 4 year starters in college).
He also doesn't need to be Drew Brees or Peyton Manning or even Phillip Rivers or Matt Shaub. The Jets have a great offensive line so he'll be protected and the running game should be at the very least effective. He also has three wide receivers that could be a #1 on many teams and a potential matchup nightmare in Dustin Keller, so he has the weapons and is in a position to succeed.
I at least hope that by your Chicago Bears comparison you acknowledge that the Jets do have a Super Bowl caliber defense. Though they do have some age problems on the defensive line Rex Ryan has proven that his schemes alone are good enough to generate a pass rush, and it's not like Bart Scott, David Harris, Calvin Pace, Kris Jenkins, Shaun Ellis and Jason Taylor are complete scrubs. With three cornerbacks able to man up on any given receiver the Jets will have so much freedom with their blitz packages.
I probably come off as a huge Jets homer, but I hope you believe me when I tell you that though I've always loved the Jets I've never had much, if any, faith in them. They have a helluva lot to prove, but more than enough talent is there, and you don't win 2 playoff games and be 30 minutes away from the Super Bowl by pure luck.