Constitutional Monarchy - Time for a Change?

Following the death of Her Queen Elizabeth and the accession to the throne of her son Charles there have been calls, in some quarters, that an hereditary monarchy, albeit a Constitutional Monarchy, is an anachronism in the 21st century and should be replaced  by an elected president. Some  calls, such as that from the Irish hobgoblins known as Jedward are frankly risible. Then there are the demented ravings from some North American harpies. Some are more reasoned and deserve some careful thought; but what I have not seen is a comprehensive proposal suggesting what alternative form of government is proposed. It is as if we simply replace the hereditary monarchy with an elected president and keep everything else the same. Yet if events of the last few days have shown us anything they have shown how the monarchy is woven into the fabric of our parliamentary democracy. You cannot simply change one part of it without having to change how the rest of it works.

There are some key questions that need answering. Do we go for something like the American system where the president is all powerful or a symbolic head of state as in Ireland?

If a symbolic head of state then how are they chosen? (Personally I would exclude any person who has held elected pubic office). If we go with an elected all powerful president is the election for her/him held at the same time as parliamentary elections? What happens if we get a Labour President and a Conservative House of Commons (or vice-versa)? If we have an elected Head of State logic would dictate that the House of Lords should be an elected body not based on hereditary peerages and the munificence of the prime minister. When would these elections be held? Should the whole of the House of Lords be elected in one fell swoop or say one third every two years? What power would they hold over the HofC?

If my memory is correct despite most Australians favouring repulicanism the last Australian referendum on who should be head of state instead of the Queen/King fell because they could not agree on an alternative. 

There will probably only be one chance to change things in my lifetime (aged 75 next birthday) so whatever is proposed needs to be well detailed and well thought out. I shall not be holding my breath