"PC pleb doesn't need data calls. All he wants to do is make a voice call to a central Police Force Control Station and be able to have calls routed to colleagues for one-on-one chats."
How can you possibly know that? What's your experience in front line policing?
"So either you go for a government funded, developed run system in a not-for-profit kind of way, or you go for open market competition and let the suppliers (more than one) fight over each other to give the cheapest deal."
That's what did happen. There was competitive procurement. Airwave won it. They're a private company but the network is publicly funded by the government. It's would be dangerous to assume that not for profit means cheaper or better - skilled RAN guys cost lots of money to employ.