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I wondered how long it would take to get to the 'Betty' bit.
How would a republic be more democratic? Parliament reigns supreme, embodied in the Queen. What would be the point of electing a purely ceremonial president? If you are going to have a president then elect an executive one, as in France, USA, etc. How many non-executive presidents can you name off the top of your head? We all know the names of the presidents of the USA, France, Russia, to name but three. Could you instantly name the presidents of Italy, Germany, Austria and other European republics?
Of course, if you elect an executive president then he will have a policy agenda which the Queen does not. Do you want a president who can overrule parliament, as in the USA? Doesn't sound very democratic to me!