It's hard for governments to implement this, because they would have to purchase Bitcoin in order to pay their employees through cryptocurrency. And then there is an issue on this process, due to the possibility of purchasing Bitcoin for an expensive price, and then on the moment they are going to execute the payments the digital currency is devalued... Then the government would face deep losses which would impact the national economy negatively.

Maybe they could create some terms where payments in Bitcoin would be only allowed when the current market prices surpass the prices they purchased Bitcoin for. This way they would never be in loss when paying their employees.

Now, about middleman payment services, I don't think it would be necessary to adopt one or another agency. The government just needs a personal digital wallet to send BTC directly to employees' wallets, like in a P2P transaction. This way the whole process is much simpler, cheaper and safer. Simpler, because it doesn't need a third party service to happen; cheaper, because exchanges' fees are abusive sometimes; safer, because it's dangerous to let your coins idle on someone else's hands, as we have seen so many scams and hacks happening along the years in crypto industry.