I have already posted my thoughts on Megxit here:
With regard to the monarchy as an institution, it's worth remembering that it has two parts: the Crown and the person of the monarch.
The Crown is the source of all authority. It is impersonal and apolitical, but gives the laws their authority and empowers the courts and the police to enforce them. The authority to make laws is given by the Crown to Parliament and only to Parliament. In theory at least, Parliament enacts the laws according to the mandate given by the voters.
The monarch is the human representative of the Crown, and exists purely to sign the papers, to physically state the intentions of Parliament via a speech and to ask the winner of an election to form a government.
Our current monarchy is hereditary because the most recent tradition it follows is the Norman-French hereditary one. But the older Anglo-Saxon and Celtic traditions required successors to be chosen from available candidates by the nobility as 'first among equals'.
The simple fact is that as long as the institution of the Crown exists, the matter of whose arse is on the throne is irrelevant.
To become a Republic would entail abolishing the Crown and creating a written Constitution. We all see the problems with that; the US struggles almost daily with the attempts to create and maintain a 21st Century state on the principles laid down in an 18th Century document.
So why not move, simply and cleanly, to a popularly-elected monarchy? OK, we'd probably be 'ruled' by a succession of football players and reality TV stars, but as long as they can sign their name and give the Speech without stumbling too often, they'll do! Or failing that, have the peers elect one of their number every five years or so. Or have Parliament select an array of 'suitable' public figures for the public to vote on. As long as they promise to refrain from political statements and campaigning for the duration of their term, it doesn't matter.
I mean, "si non confectus, non reficiat", right? So don't change the system, just the people,