Actually, visit their website and you gonna see that they indeed do just that.
You are quite correct, so they do. Perhaps I ought to have written "should call itself the UAE's National Airline." I suppose the deal under which Abu Dhabi bailed out Dubai from bankruptcy a few years ago may have included nominating Abu Dhabi's airline as the National Carrier, along with calling the tallest building Burg al Khalifa, instead of its planned name. But I very much doubt that the other Emirates regard a certain airline based in the West of the UAE as their National Carrier!
Funnily enough a private airline started in Abu Dhabi in the 1970's called Emirates Air Services or something very similar, which took a couple of oil company contracts off Gulf Air, operating DH Dash 7s, but it sank without trace.
Prior to that, in 1967, Sh Zaid issued a decree to an Englishman to set up Abu Dhabi Air Services as Abu Dhabi's National Carrier, as a limited company with Abu Dhabi shareholders. But Gulf Aviation, now Gulf Air, and BOAC stepped in very quickly and killed it at birth by persuading Sh Zaid to buy half of BOAC's 50% share of Gulf Aviation. Bahrain and Qatar already had 25% each, Abu Dhabi bought 25%, and the remaining 25% was earmarked for, and bought by Oman. The dream was an immensely powerful airline owned by all 4 States, operating as the National Carrier of each.
But Dubai refused to accept the Abu Dhabi's share was the UAE's share, and within a few years Emirates was born. That triggered to start of new airlines in Qatar, Oman and of course Abu Dhabi eventually, and poor old Gulf Air reverted to being Bahrain's airline only.
The name Abu Dhabi Air Services was eventually used when the existing two private handling agencies were bought by the Government and merged into ADAS, in the 1970's.
How did all that potted history come from a story about turbulence?!