The role of the British monarch is more complex than most may think, even the Brits themselvers in many cases. No law of Parlaiment may become effective without the monarch's approval. Unlike a bill sent back by the House of Lords, the Commons cannot overide or ignore a bill sent back to them [and it is stressed that this would only be done on the advice of her ministers]. All acts of government are only legitimate with the approval of the sovereign.
Another common misconception is that the succession is automatic. It is not. Only an act of Parlaiment passes the throne from one monarch to the next. This is now pretty much pro forma, automatic, as the Act of Succession is not really contested, but the myth of "The King is dead, long live the king" is just that- a myth. Parlaiment can bar an inheritor, if it so choose to, and risk the dire consequences. The same is true for abdication. A monarch cannot abdicate of his/her own free will. It takes an act of Parlaiment. Even if the present queen were incapacitated, it is highluy unlikely abdication would take place. A regency would be much more likely.
Also, the constitutional function of the sovereign may come into place with the total breakdown of Parlaiment- that is, the inability to come to compromise or coalition between the parties. The monarch may disolve the House, appoint a "caretaker" PM and call new elections. Although in the UK, so far, this has not been called for, as the various parties do reach agreements, in other countries, it has proven vital to the continued running of the country. Namely, Belgium, Netherlands and Denmark. In European republics, the president, elected or otherwise would preform this stabilising function.
There are a lot more subtleties in the running of the country, just everyday life, that are in one way or another directly linked to the monarchy. [NOT necessarily the monarch herself] and it could be dismatled. We shall see just how much so if the move towards Scottish independance goes, but the process would be very cumbersome and indeed costly.
But personally, from my own experience, I do not see much enthusiasm for such drastic action, at least amongst my English friends.