Today is the seventieth anniversary of the Abdication of Edward VIII (unless you're in the Republic of Ireland, where it's tomorrow!). It was just the latest in a long line of actions that have shown that the monarchy can be flexible and overcome particularly problematic individuals that dates back to at least King Stephen and also includes King John, Henry IV, Henry VII, William III and Mary II and George I. The monarchy rapidly recovered (not least because of George VI's stand during the war) and it's only recently that the issues of the Abdication have become relevant once more.
It's often said that one of the basic points of a monarchy is that there is no choice whatsoever about the succession. But this overlooks a vast amount of precedent of both individuals and lines being passed over. Were one to follow a notion of strict male preference primogentiure then our current monarch would not be Elizabeth II but Francis II. Time and again a way has been found around a problematic individual, allowing the system on monarchy to survive.
Contrast this with France in the nineteenth century, where it wound up with a restoration putting on the throne die hards like Charles X. A legitimist succession to the Orleanist line (and ignore the Carlist line, as most legitimists did) before 1830 may have given the monarchy the stability and broader support it needed to survive, but ultimately the monarchists wound up supporting Henri, comte de Chambord, who made impossible conditions for taking the throne and ultimately wound up being the one man who made a France a permanent republic.
I rather doubt that Prince Charles will wind up being the UK's come de Chambord. But it's hard to deny that his accession is creating uncertainty. His marriage to Camilla is exactly what his great uncle was officially forced to give up the throne for (and wasn't "Duchess of Cornwall" precisely the title proposed for a non-Queen Wallis?). Now social attitudes have changed a lot in seventy years but the prospect of a Head of the Church of England being married to a divorcee is one that many find hard to accept. Traditionally the monarchy and the Established Church have reinforced one another. Could we see one bring the other down? And which?
Then there's the issue of public popularity. Whilst the newspapers seem to have restrained themselves in recent years in reporting the "Royal Soap Opera" (give or take the Daily Express obsession with Diana) the media does still retain the ability to break a public figure's credibility. Charles has fought a long battle for media, and thus public, approval and in many ways its not over yet. Can he successfully reign if he's constantly having to work like a politician to maintain popularity, in a job that is defined as not open to politicians?
But equally a lot of this would apply to William (although he also has to navigate the problems that the title "King William" will bring). A public debate and choice of "Charles or William" amounts to a limited choice republic and it would be hard to resist the logical consequence. Will the monarchy be able to weather a second storm? And what will happen to the Church of England?