Would appreciate it if some of her subjects could enlighten me as to the merits of a Monarch for the U.K.
I'm still Her Majesty's subject though a long-time resident of Texas.
I don't have any good answers to the questions asked by the OP. (This is my first post so be easy on me
.) My best guesses would be that (a) the Queen and the royal family are considered entertaining, harmless, and a generator of tourist revenues by most UK citizens and (b) a minority of people, such as myself, who think it's wrong to appoint people on a hereditary basis, are mostly not so worked up about it as to do anything about it because we see the role, in practice, as just ceremonial.
To me, the monarchy is distasteful and useless but not positively harmful. I'd still like to get rid of it. I wouldn't want to lumber the PM with more ceremonial duties so I would favor an elected figurehead (or a somehow appointed figurehead) to replace the monarch.
I don't think the monarchy provides anything by way of checks and balances. Although the monarch may have some theoretical role in restraining the excesses of the parliamentary government, I don't think that role has been exercised in hundreds of years and if it were to be exercised today, the government would probably respond by abolishing the monarchy.
if one thinks that more checks and balances are needed, I think that one would have to look to reforming the House of Commons, the House of Lords, and the judiciary. Maybe a written constitution would be helpful. Abolishing the monarchy would, in itself, in no way lead to destabilization of the country since the monarch invariably just "rubber stamps" the decisions of the government.