I was surprised that England has never had its own official anthem considering it was an independent country for about 800 years, but actually I think you're right.
However, my point still holds - GSTK was originally a song in honour of the English monarch rather than British (as it was written long before the creation of the Kingdom of Britain, which later became the UK). It also became established as the UK's national anthem by virtue of it being sung in London theatres in support of the English King George II who was in battle with Bonnie Prince Charlie at the time. Hence the lines added in that era "And like a torrent rush, Rebellious Scots to crush, God save the King". If you're interested, more information officially sanctioned by her majesty here.
Leaving aside all of that anyway, it's an anthem that at best only about 50-60% of the population of NI identify with, and even then they identify with it in the context of being "British" rather than "Northern Irish".
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