It's a fair question, and I am not ideologically a monarchist at all (quite the opposite really) so I don't dispute for a second that life would continue 99.9% the same as it is now. I could argue that on this, I support the status quo, so it is for others to make the case for changing something, but nontheless, a few thoughts;
1 / Global exposure. I've been struck by this - it's led the news websites, rolling news and papers in many countries for days. Some may argue that is not positive, but it is great advertising for the country and culture. Edinburgh and Scotland in particularly looked glorious. I think having a reigning monarch helps maintain that and is a significant part of the tourist offer. Cardiff will be in the worlds eye on Friday. We cannot buy that level of exposure.
The USA had global exposure with Trump and Russia has global exposure with Putin. It doesn't really mean anything per se.
2 / Stability. The UK is actually a pretty stable country and we have dealt with constitutional change in a generally stable manner over the last couple of centuries. I do wonder if an apolitical head of state does help with that. I'd suggest that perhaps it does
The UK is more unstable than some neighbouring countries that do not have royal families that are worshipped a la House Windsor - and the UK seems far more volatile than some of those countries.
3 / Tradition. I'm not saying we live our lives for tradition alone, but it does matter. What else is it about Cardiff City playing in blue other than tradition? It's not a tradition I greatly care for, but many do, and I think I would miss elements of it if we had an elected politician as head of state
Bear-baiting and sending children up chimneys was traditional - and not being able to get red of someone who achieved their position merely by being born in the right bad is ridiculous.
4 / It aint broke. The new King is powerless. He has now control over laws that impact us. If he did I would think 100% differently, but he doesn't. In a practical sense, removing him wouldnt lead to any better laws. It may even lead to worse if we had a president and prime minister from the same party? I've seen more people on here this week talk about the Queen than they ever do the Welsh health or education system - these are things we can and should influence far more.
The elitist and class system in this country is indeed broke. Hereditary monarchy, hereditary peers, the political dominance of those who had private schooling and/or attended one or two universities, the sleazy honours system, the nepotism, the still-existing class divide etc
5 / Kate Middleton is an absolute megababe and the thought of her as queen is hot.
More than anything, I just don't see the point. We know it doesn't make enormous practical sense, but nor does religion for most of us, but sometimes you take that away from a society and a void fills it that may be less positive than what it replaced.
Both royalty and religion are infanitising. Let's grow up and realise we need neither.
And thats not to say that there aren't a great many things the Royal Family could do better, because there is. In the absence of any Republic being imminent, perhaps that would be a more useful conversation?