I love that last sentence, but then what if the monarch is also barking mad? The public never even had the chance to vote them in at all.
We'd have to be really unlucky for them both to go bonkers at the same time.

You're correct to specify the things that the royal family does. The argument on that point really is whether or not this is the job for a publically funded hereditary line of people in whom the public have absolutely no choice over.
Fundamentally since you said you're all for the royals, do you have any problem at all that in a democratic society there are people born into such a position of power, prestige, wealth and imposed respect? (I'm aware that capitalism naturally produces wealthy lineage elsewhere but I'm referring to a publically funded family with certain jurisdictions over the whole nation).
I have no problem with monarchy being "publically funded" (they pay taxes now, so they are paying partly for themselves), the monarchy is what sets us apart from the scruffy little urchins such as France, Germany, Italy etc and the upstarts like US of A. They bond us by tradition, trade etc to the Commonwealth states
I love that last sentence, but then what if the monarch is also barking mad? The public never even had the chance to vote them in at all.