Presidents are not necessarily the ideal solution as to what to do about having a symbolic head of state. Moreover they don't come for free or without controversy (and that, every X number of years - whatever the term of the presidency).
It is very difficult in the British system to clearly determine what is, while theoretically a personal prerogative of the monarch, practically impossible from those personal prerogatives that have simply not been exercised recently, perhaps in part because the possibility of them being invoked constrains the government to work with the monarch in a way which will not result in them being used.
Can not or does not?