This isn't entirely true. Charter operators do not typically file their flights as their registration number here in the US. They will usually file as airline code + flight number just Large charter operators are, in fact, airlines (certificated air carriers if you'd like to get specific) and they all have FAA radio callsigns just like airlines. A few examples off the top of my head:

NetJets = radio callsign "Execjet" (airline code EJA)
Delta Private Jets = "Elitejet" (ELJ)
Flight Options = "Options" (OPT)
Solairus = "Twilight" (TWY)
Flexjet = "Flexjet" (LXJ)
XOJet = "Xojet" (XOJ)

And even some Part 91 GA operators use airline callsigns too. I've flown on GLF flight numbers before (radio callsign "Gulfstream Test")