IMO these days the notion of a "flag carrier" is somewhat outdated and very last century.
Consider the "good old days" when just about every euro nation had to have their own airline flying their people to all points in the world, even if it made very little sense (eg: SABENA, JAT and so on) but was ALL about national pride and in some ways geopolitical - ie: if we can fly you from XYZ home country to JFK, LHR etc then we're on the map as a "somebody"
This even extended to Asia where this notion still survives to some extent (eg TG is a bit of a basket case but still gives it a red hot go).. MH I think finally gave in by ditching F but still plying the skies to/from LHR, but even they saw sense in swapping 380's for 359's. And, it was more or less nationalism that saq SQ split from the original MH (and yes, they became a much bigger and better carrier) and, outside of the ME3 are probably the biggest real "flag carrier" left given the size of Singapore).
In the USA, PA was the undisputed flag carrier for decades until it went under.. these days there's no real "flag carrier" any more even with endless consolodation to "the big three" of UA DL and AA (plus all the rest).
These days both with the failing of the legacy carriers in smaller places (the aforementioned SABENA for example, MALEV, Aer Lingus in it's various incarnations, Olympic.. heck even Swiss born of Swissair) and the rise of powerhouse LCC's in europe (Think Ryanair, EasyJet, Wizz etc) t and the establishment of the EU as a single market, plus domination of the big players LH, KL-AF and now IAG(BA,IB, EI etc) your flag carriers as such are far less relevant (imo) then they once were.
And yes, even Australia where QF, which was very much akin to PA for a long time holds the notional "National Carrier" moniker more through history than anything else. IMO it's the only reason they hang on to LHR and NYC services (and will offer F on project sunrise allegedly).
And let's face it, VA could hardly call themselves a flag carrier given their very limited international network. True QF have paired back a heck of a lot from the heyday, but they can at least claim to serve all major populated continents (obviously JNB and SCL are really stretching it for own metal but at least they do
).
These days though, for most carriers in the world, it's about partnerships and codeshare as much as it is about the reach of one's own metal.
my thirteen cents worth