Hmm I feel like it would be difficult for a private company to get approved to take over the TSAs job.
You misunderstand. It would still be TSA staff, however they would be dedicated to Clear members. Who would pay for it would likely be Clear.
Delta does this already with their SkyPriority service. It is a separate security line specifically for Delta First class (or Skypriority member) passengers. No amount of Clear, TSA Pre will get you into that dedicated line.