A few points need making.
1. The Royal Family is hugely profitable. We pay them not that much money and get a vast amount of tourism in return. Buckingham Palace etc wouldn't be as big a pull if it was just a big house. In London this week we could probably have paid for the wedding just from the cash spent by American TV networks transplanting their entire operation to Trafalgar Square and Green Park.
2. The monarchy has no power. A lot of powers exist as something called "crown perogative". All of which are exercised by the Prime Minister and/or government. technically the monarch can do many things - declare war, dissolve parliament etc. And if she tried to do any of those things an Emergency Powers Act would go through Parliament, she'd resign, and the powers would be formally taken away
3. Its not as if a Presidency is a good idea anyway. Most Brits skin crawl at the idea of a President Thatcher/Blair (delete as appropriate). And from this side of the pond we've watch with great amusement as you elect first a womaniser who fracks anything that walks, then a cretin who half the country thinks is a moron, then a man the other half think is a commienazi foreign muslim.
We have a Queen. She is 85, and with her 89 year old husband still have to spend large chunks of the year consulting the PM signing bills being consulted doing state visits and pretending to be interested at an age where most people are either dead or doubly incontinent.