GF or its replacement will be a reality and it won't be just private. It cannot be yet.
Proud as they are in this part of the world for now(?), they may choose to change the name to Air Bahrain or Bahrain Air(ways) or whatever... It isn't important for now, but Gulf Air (GF) is a trade mark and it may be sold for some money to a private investor for some money while the Bahraini government chooses to have a new name to portray the country name internationally. All is possible.
I would bet that GF or the new airline would be something like 50.01 and 49.99 if you like. The Bahraini government will have to have the majority stake. It's too early for them (government) to let it go and become a minority partner or play no part at all.
Again and with whatever name, they will have to rely on the local expatriate workforce and cargo services. The so-called through traffic will add to it if can compete with the other prices (QR, EY and EK besides the LLCs). They will also have to diversify and get holiday packages, but again, it's going to be difficult to compete with the Gulf majors. Perhaps, becoming more sophisticated and offer "excellence" in first and business classes may help to create good revenue and its own niche. A guess professional "connoisseurs" of the marketing and advertising will have a better plan. It has to be someone with solid background and expert in the local market needs.
Needless to say that, and without being arrogant, Dinars and senses will have to be the rule. The government has to give the new management team a sensible and viable mandate to achieve pre-determined goals in (let's say) 3 years without going mad to show "convenient" results as it happened recently despite the difficulties created by the countries involved.
Corruption in whatever form and from whatever quarter will endanger any (recovery or otherwise) plan. The government, potential private investors, the board of directors, management and their associates will have to be more responsible and accountable. Without this attitude from the top towards the bottom, nothing will ever change.
20 twins (10 narrow and another 10 wide-bodied) would probably be a reasonable (re)starting line-up.