Short answer: No.
I am not the ideal person to put the case for preserving the Monarchy ... I find it ridiculous that our head of state is selected by being born to the right family, and I certainly would not choose a monarch to rule a new country. And I haven't even started on making a point out of the fact that our Monarch's true surname is Saxe-Coberg-Gothe-Himmel or something or other (i.e. they're really German). But it does have some advantages.
Some of these advantages are relatively small, but they all add up.
Firstly, whenever a state comes up with some method of choosing a head of state which does not involve finding the person born with the biggest silver spoon in the mouth, they always choose a politician. And making policticans more important is something I regard with horror. So do most people who are not politicians themselves.
Next, the queen knows that if she exercises her power too much or too often she (or her family) will be demoted. Despite what people might think, the monarch does have quite a bit of real power that is simply not used. However that power can be used in an emergency, and even the threat of using that power can cause politicians to rethink their excesses (who knows how often she has persuaded a politician to think again?).
Of course even influencing a democratic government is never a good thing, but on some occasions may be better than the alternative. It may be that a president would have more 'right' to perform this function but a political president would be tempted to excercise his or her power in a way that a monarch would not. There are plenty of examples where presidents have become a little too powerful and the country involved becomes a little (or a lot) less democratic.
If it isn't broken, don't fix it ... why mess with what is a stable and more or less successful form of government ? The chances are that if we abolished the monarchy and became a republic that we would still have a stable government, but there is still a real risk that we could have problems. Stay with what we have!
Finally of course, the monarchy is a great tourist attraction that brings a certain amount of tourist spending to the country. It is doubtful that a president would perform this function.