SASless:
If a company can provide SAR cheaper and make a profit, then surely there is scope for a Government Agency to do it for a lesser price, i.e without profit, hence even cheaper for the taxpayer. Unlikely, I know, but theoretically possible.
The CG would never change their contract as they have no jurisdiction above the high-tide mark. They want to keep their helicopters for themselves and any "sharing" would only reduce their cover. As soon as a helicopter is tasked inland, there's nobody to cover the sea (because there is usually only helicopter at each base and certainly no seconds crews).
If a tender for SAR cover was offered which required an equal standard to that provided by the military now, it would take any civvie company a long time to train crews and adapt helicopters to meet the same overland standards of today's military. The CAA would have to become inherently involved and permit special AOCs, which although I know they would do, they would need a lot of persuading. So, I admit, nothing is impossible. But is anybody willing to take the risk?
Now, lets look 10 years into the future. The military expertise has all left (to the civvie world). The civvies have to actually train new crews now as their supply has dried up. There is suddenly no more competition to the civvies and hey, guess what, the price for the service, which the government feels obliged to provide goes through the roof. How will the taxpayer feel now, with, no doubt, the profit margin increasing every time the contract is re-signed.
You are right, competition never hurts. A financial level playing ground would obviously show differences in cost, but when would the standards of civvie overland SAR meet those of the military on a level par? - years?, decades?
I still think the Govt Flying Service is still the best option and I'm wiling to fight my case. I can go on, but I'll stop here and see what you all think.