The reason the government doesn't do that is talented people don't want to work for them. Virtually all the government jobs are union and pay is primarily based on time served. You'd be getting third rate talent at best.
I don't think that's the real reason, to be honest. More like some combination of anti-hiring ideology and a handful of people getting rich off the process. The British government did it with, apparently, real success. And are government contractors really much better places to work?
No instead they hire contractors for that purpose.