This is not true.
In late 2014 47.8 per cent of QF was foreign owned. (by looking at the
Home - Australian Securities Exchange - ASX or others, a more recent figure could be sourced.) It is restricted under the Qantas Sale Act to 49 per cent foreign ownership.
It is difficult to take QFi all that seriously on a number of its routes because it does not (for example) usually fly nonstop daily SYD to PVG, SYD to MNL, SYD to CGK, MEL to HKG, MEL to SIN or at all from anywhere in Oz to KUL using its own metal.
It survives because many corporates and government agencies use corporate travel agents who book it first and who seem not to consider alternatives. As another on AFF remarked, QF (and JQ) have been lucky regarding safety incidents. Notwithstanding that the aviation industry worldwide (regular public transport, not single engine joyflights) is undeniably safer than even five years ago, the law of averages suggests that eventually - who knows when - JQ and QF's 'luck' may run out. Not that I'm wishing it on anyone, of course.
QF's share of leisure travellers who pay for their fares out of their own pockets must be less than 20 per cent on international routes to and from Oz.
Even on routes where it runs daily, QF's frequency is poor or mostly less than its competitors. It still considers SYD as its hub: despite Melbourne and Victoria gaining 100,000 net new residents every year and Melbourne now being not much smaller than Sydney, not many QF flights run nonstop from or to Melbourne internationally.
And yes, the company is unionised, staff are promoted on a seniority basis rather than on merit as far as I know and some that I have come across are unfortunately arrogant.
There is little cause for arrogance when three out of four travellers flying internationally to or from Australia choose a competitor rather than QF or JQ. Even if EK is included, two out of every three of us choose an airline other than EK, JQ or QF, and that is with artificial maximum capacity restrictions ('permitted annual seating capacity') on a number of routes.
QR would like to fly to more than MEL and PER but cannot at present. 5J would like to fly to MEL and only now can it consider doing so as the number of seats on that MNL - Oz route has just been increased. CX would similarly like to increase its flights into and out of Oz but if I recall has reached its ceiling.