I see, but is the Monarchy crucial to the existence of government, and if not, do they at least contribute to the overall welfare of the UK and why?
yes they do the entire english tourism industry is based on their royal family
also the UK is the most important country when it comes to the development of constitutional and parliamentary law, the concept of a modern parliament was built in GB and that alone is enough not to replace the pm with a president
the us parliament is mostly based on the uk parliament and the president is only one person who wields such power because before america had a president they had a british king and later based the president on that
also the us constitution wouldnt exist without the development of constitutional law in britain, altough britain itself has no constitution
i could write much much more and much more accurately, but whats the point?
PS: if the UK became a republic they wouldnt have a president like potus they would have a head of the state with severely limited executive power and a pm with the same authority as he does now
also the british queen has a german bloodline

kinda funny isnt it?