The Head of State in the UK is not just ceremonial.
Ask any member of the UK Armed Forces about this.
The ruling monarch is their commander-in-chief and also the representative of the state to which they owe their alliegence.
You may not believe it but they do!
Also note the American system of government consisting of Head of State(also C-in-C)/Upper House/Lower House is quite strongly based on the British model.
NB At the time of G. Washington there was significant discussion as to the form of address to be used for the new president. (Should it be Sir or Sire ?, or some such argument.) Thankfully Mr Washington chose to avoid kingship :)
AndyG