They likely didn’t remix anything on The Hunting Party. They do have some unreleased remixes, like an unreleased Castle Of Glass remix used in a 2013 LPTV (not the Recharged one) and a bunch of Meteora remixes they never put out.
Any attempt by Linkin Park to go into a studio to “reimagine” a song like they did with Reanimation would likely yield great results. They nailed that project. That was taking the original song and “reimagining” it to a completely new extent and then featuring guests/friends on it. Recharged was simply electronic remixes of Living Things. Two different projects entirely. There’s no doubt that Reanimation was a much superior project.
And yes, Linkin Park wouldn’t be where they are in their career right now if it weren’t for Rick Rubin. He completely changed their process in how they write songs. It’s good he’s gone now as the band can definitely produce their own efforts, but his work was appreciated for sure.