This is a hobbyhorse I have ridden before!
I think that 'flag carrier'
does mean a state airline and not a large commercial company. As such, the UK has not had one for a long time and Europe has moved beyond this fashion. This idea has always meant more to the politicians and those who believe in their country being the best. I always rail at those who suggest that a 'plc' is a flag carrier.
It is my view that the younger generations (under 45/50 as an arbitrary line) are only interested in the price and service - not who owns the company.
In short - it makes no never mind to me. I look at their safety record and all the usual factors before booking, not who owns them. I view the insistence of BA that they are a flag carrier as simple marketing. Likewise when VS painted flags on the winglets. The image is meaningless to me but if it gets them some customers then it's worth the paint.
Countries used to like have a flag carrying ocean liner but most of them were private and then they 'flagged out'. Flags were fashion then and now.