Question: Why does Britain bless the Queen and not the people and the country?
Answer: Tradition, or inertia--the anthem's official status, as well as presumably its lyrics, derive from custom and usage not Royal Proclamation or an Act of Parliament.
An English audience first asked God to save the King in 1745 as a show of public support for George II after his army's defeat at Prestonpans at the hands of the Jacobite pretender, Bonnie Prince Charles. While Wales has its own national anthem, and Scotland has two of them ('Scotland the Brave" and "Flower of Scotland"), there's a movement afoot to establish the English national anthem as either Blake's "Jerusalem" or Elgin's "Land of Hope and Glory".
Since several attempts have also been made to revise the current UK national anthem's wording--both to include all parts of the UK as well as to put the emphasis upon the UK's citizens rather than its monarch--perhaps a revision similar to the second version of Canada's official second version of "God Save the Queen" would work:
Our loved Dominion bless
With peace and happiness
From shore to shore;
And let our Empire be
United, loyal, and free
True to herself and thee
For ever more.
Of course, the melody to "God Save the Queen" is also often sung to the words of the United States' patriotic hymn, "My Country Ti's of Thee":
My country ti's of Thee
Sweet land of liberty,
Of Thee I sing.
Land where my fathers' died
Land of the Pilgrim's Pride,
From every mountain side,
Let freedom ring.
BTW, we Yanks are always happy to help out the Mother Country as well as our long time allies--Canada, New Zealand, and Australia. In the meanwhile, be of good cheer. At least, "God Save the Queen" is easier to sing than "The Star Spangled Banner".