Well, the attraction isn’t “Royalty”, it’s the function of that office in a modern democracy that’s the issue. If it doesn’t work, throw it out or ammend it. It’s not exactly sacred. And it’s not about “cute” castles, either.
For me though, it means the Head of State is not party political, (under the present system of funding) is ring-fenced from corporate influence and takes the long view, the one beyond the 4 or 5 years of the elected and self-serving Government.
Probably the most significant aspect – an indirect benefit, almost – is that it permits the Executive (through Parliament) to be surprisingly dynamic, no need for ‘deal-making’, compromising to the point of redundancy, etc. – in that sense, it’s what the whole shape of ‘Government’ (in the wider sense) allows, of which the Monarchy is a component.
I also think there’s a relevant ‘continuity’ issue, I always have trouble putting my finger on that – but somehow, it’s useful that the present head of State learned her trade when Churchill was PM.
And I’m not overly keen on the military owing allegiance to a (by definition) transient, self-interested politician, either.
My main concern is that we don’t know how, in a modern setting, the dynamic/Constitutional checks and balances between the Monarch, the Government, Parliament, the media and the people works in a genuine crisis. For example, if the PM pushes his/her mandate too far.
In theory, it’s a great system (in a Heath-Robinson fashion), but we don’t actually know for sure.
The cost, for me, is irrelevant - I just want the best system possible and anything less is a false economy.