Sure, they could have and should have tried to run it with Marshawn Lynch, but the fact remains that Wilson was to blame for the interception, as any White QB would be.
I know this is unpopular now and even then, but I disagree about running Lynch. Heck, I remember getting into huge arguments defending the Seahawks for passing the day after the Super Bowl.
There are a few factors why.
1) So called Beast Mode Lynch had not been good running the ball from goal line situations especially from the 1 that year. He was 1 for 10 on those.
2)If you want to criticize the Blackhawks for anything, it's for assuming Belichick would call a timeout after Lynch got stopped at the 1. Because of that assumption they wasted too much time to get the next play off.
3) Because they didn't consider that possibility, they didn't have a proper play set up and if they failed running it in on 3rd down, they was a possibility of not even getting 4th down off.
4) Which made the offense predictable on 3rd, forcing them to pass to have one more play in case it failed.
5) And with Wilson not throwing the best of balls, the interception happened.
These are the factors on why they lost there.
That and according to the first NFL Film "Do your Job" documentary, the Patriots knew through film study that the Blackhawks ran the specific play and practiced defending it during the season.