I could see it having success but it depends on what the airport is willing to pay for. In Florida you need a guard license to work as a security guard in any capacity, which means you have to pass a state exam and have a certain level of training by certified trainers. The more requirements for the job, the higher level of training required and of course a higher expectation for the wages. Given they say they would use the federal guidelines as their base requirement I would expect that would mean a high level of training. If the airport is willing to pay for the right kind of officer, they could be very successful.
One advantage is quality control over the screener crew. In short, the airport manager could easily remove someone who wasn't performing to standard. For example, if a screener wasn't performing good customer service, the airport could have him/her removed from the site immediately. You don't have that level of quality control with TSA.
Private security firms already provide airport security services at airports around the world with a high level of success. Private security firms already provide a great deal of security at US airports, but they are behind the scenes or in cargo areas so no one sees them. They also do security at hospitals, stadiums and many critical infrastructure sites.
In my view, we have 2 viable solutions. First, if we think the threat is indeed serious enough, then we pull the TSA screeners into the realm of law enforcement. We hire and train them to be federal law enforcement officers, give them authority and the accountability that goes along with that authority and make them do the job right. If we don't think the threat is that serious (and my view is we don't) then we allow the airports to do their due diligence, assess the risk based on federal standards and let them hire private companies to do the job. TSA is too much of an in-between element, appearance of authority without any and appearance of importance beyond reality for them to be truly effective.