That's exactly the trick. Have the head of the state encrusted in titles, ceremony and tradition, but with no practical power. Leave the gritty parts of politics to the parliament. Let the King shake hands and open buildings and leave the governing to the MPs. It was basically figured out over centuries of compromise and finalised with William III's Glorious Revolution. Britain has had some bad eggs but never a tyrant. We've never had a Robespierre, Stalin, Mau or Franco. It seems to have worked.
> Save money
I look forward to finding a politician who will work for nothing. The royal family pay for themselves through their own property and holdings, they actually pay 80% of their earnings to the state and keep the remaining 20% for themselves.
While how much money is spent on American presidential elections? It will frighten you. That money has to buy something, right?
> Nowadays, what's the point? Why do we need to continue with that?
Generally we continue with things because they've worked well for us for centuries. I don't see anyone in the current political class that I'd rather have as a 'president' than Elizabeth II or even Charles III. The British way is empiricism, biased more to observation than pure reason.
> I just would like to hear real arguments about why a monarchy is so much better than an alternative.
It gives us someone who can speak for the nation, untainted by party politics. This is something you'll miss if you abolish the monarchy.
If you are actually curious to read the other side, I'd recommend Roger Scruton.