The Queen is just the head of state but doesn't have any actual power unlike a President. How would we feel if that psycho from across the pond was our head of state, with his finger on the button?
Ah ... the old misconception.
When people think "president" they always look at the USA and think that its constitutional arrangement is the only one available. The USA has an executive president: the American president acts as both national figurehead and working politician. There are countries where this model has not been adopted, Germany, for example.
The presidency of Germany is an honour given to someone who has brought credit to the country.
Why not allow someone who has received public approbation for his or her achievements and the credit they have brought to the country as a temporary head of state not just someone who is the result of a random act of sexual intercourse? Certainly, the present monarch, like her father, has been an exemplary figurehead. But her uncle wasn't, and, had he not panted after a totally unsuitable spouse, we might have been stuck with him and his inadequacies for 36 years.
There are people around whose occupancy of a British presidency would be widely admired. I'll throw a couple hats into the ring: David Attenborough and Stephen Hawking. Perhaps even Mrs Elizabeth Mountbatten.