Airbanda
You could add Northwest & Braniff to the list (and others serving short haul overseas routes).
Pan Am used to be considered as the chosen instrument although Juan Trippe lost a lot of influence in later years.
The US limits foreign ownership to 25%, one of the arguments being that the airline might be needed as part of the defence effort. There is always the Fly America programme but that has rather been overtaken by code sharing.
What has changed is the reduction in state owned flag carriers. Some flag carriers were partially privately owned, Swissair being an example. SAS is interesting. The Governments of Sweden Denmark & Norway owned 50% and the other half was privately owned. Was SAS a flag carrier?
What is interesting is whether a state needs a flag carrier. If, for the sake of argument, Alitalia ceases trading, other airlines will obviously move in to fill the gap. However this may only be increased feed to their own hubs. This would effectively de-hub Milan & Rome and reduce the number of non stop flights and destinations served, particularly long haul, which might make the cities less attractive as a international business centres. Exponents of airport expansion would certainly cite this although you might argue that Geneva, for example, wasn't hurt by Swissair going bust (although SR had largely de-hubbed there at the time of its demise).
The EU believes that countries do not need flag carriers. However it could well be that BA, LH, AF and possibly KL are too large to fail. It would certainly be interesting times if any of them were to become insolvent (and AFKLM or at least the AF side, is currently looking frail).