Having an inert (a-political) monarch provides a figurehead and commander of the nations military, making it less likely that coups and fascist takeover occurs because all the soldiers swear an allegiance to the king. They don't have to agree with him, or even with monarchy, but as the big boss ultimately in command of the entire army, an allegiance to him is allegiance to protecting all your comrades and countryfolk.

The monarch is usually unlikely to collapse the country and then flee to another one with their winnings, as a prime minister might do. You could say they have commitment to it, since it's essentially their family business. Theoretically it should be a blessing that, in a monarchy, we have leadership who actually inherit that position and are raised from birth to lead.

One reason people don't like monarchy is because there are past case studies of it causing problems, e.g just one legitimacy crisis and civil war breaks out, or if a monarch gets to greedy they will emnroil you in war with your nearest neighbouring state for ages.

But people need to realise, it was never just the monarch in charge. There exist pressure groups, the aristocracy, the church, merchant class; these sorts of organisations have a lot of say in when a war happens and what gets done. The truth of power is oligarchy - an organised minority of the population will always be best suited to maneuveur a takeover of the rest of the popupation. It has to be larger than 1 person, though.

In my eyes, therefore, the king exists not as the leadership but as the figurehead of the organisational force of the country. For instance in the UK the civil service works for the king - this is again superior to them working for the party government, as they have more reason to be unbiased than just "please don't do that, it wouldn't be fair." The monarchy is a real thing, bigger/more important than the party politics, even if less flavourful.

I wholeheartedly also believe that the British Crown provides superior organisation of our secret service/intelligence agency (nope, its not just movie-slop!) - we actually have the oldest standing secret service of any country, i believe, and it's pretty impressive russia hasn't completely destabilised britain yet. I would chalk that up to the crown.

The british monarchy is all of these things, but for countries across the world. It therefore has staying power. Getting rid of it isn't merely a question of how would the people of the UK feel - it may lead to little crises across the world. It's a big reason why the british army is called out to fight wars in african nations (they do this a lot btw)