England does not have a monarchy. It is not a separate sovereign country. The United Kingdom has a monarchy.
The point is that any country needs a head of state. This can either be the same person as the head of government, as it is in the USA, or a separate person, as it is in most countries. If it is a separate person, they have mostly a ceremonial and figurehead role, including appointing the head of government (usually called the Prime Minister), signing Bills into law, and formally calling elections. This can be either a president or a monarch.
If you have a parliamentary system, as these countries do, you need someone to actually appoint the Prime Minister after an election. Normally it's obvious who that should be, but the appointment still has to be made. And if a separate person is responsible for deciding whether to agree to have an election, it's possible to be more flexible and not need to have elections on fixed dates, because the power to call an election isn't in the hands of one person. So while there aren't many things the head of state HAS to do, those few things are vital and you need them to exist.
Generally in European countries there is a ceremonial president of this kind, though you've probably never heard of them. (France is an obvious exception - the President and the Prime Minister actually share power in France, but that is a less common system.) Try naming the German President (and don't say Angela Merkel because it's not her - she's the separate head of government, the Chancellor). Now as this post isn't responsible for making any real political decisions, it works just as well if it is held by an hereditary monarch. And as the UK already has one, why bother changing it? It saves the cost of another regular election, too!
What you tend to find with a ceremonial president of this kind is that it's some washed-up politician. Actually live in a constitutional monarchy like the UK and you find people tend to like it, because the Queen is totally non-political and can be truly the head of the nation for everyone. That's the real point of it. The vast majority of us in the UK can see her as someone to unite around, and she understands this and is careful never to say anything controversial. Which is why she never gives interviews. For an example of another European monarchy, the Netherlands celebrates the fact on King's Day every year and that's a great festival of just being Dutch. The British armed forces swear their oath of allegiance to "the Queen, her heirs and successors" and that's so meaningful to be able to swear loyalty to a real person.
OK, the UK could do away with the monarchy. But if it did, it would have to totally rewrite the constitution, or the easy way to do it is just replace the Queen with a ceremonial president. Do we really want to replace her with some washed-up politician? Nah, not really.
Her Majesty does quite a bit behind the scenes, though. British Prime Ministers like this - the PM meets her every week and they like being able to talk things through with someone who has also seen all the secret papers but will never, ever, leak what they said.
And having a separate head of state and head of government is, as I said, more flexible. An example of this happened in Australia in 1975. The two Houses of Parliament were opposed over the Budget and the country was heading for government shutdown. To avoid this, the Governor-General (who the Queen appoints to represent her in Australia), sacked the Prime Minister and appointed the leader of the Opposition on condition that he call for an immediate general election. Of course he did as that was the whole point of him engineering the situation in the first place, they had the election, problem solved. Her Majesty can be a handy tactical nuclear weapon sometimes! Of course the same could be done with a ceremonial president, but why change to have one?