What exactly is the justification for having those tasks, if they're deemed necessary, executed by some hereditary, secretive, uber-wealthy family rather than by someone who is actually a civil servant?
The easiest ways to corrupt someone are offering them wealth, opportunities for their children, or a popular reputation; the royals are relatively immune to all of those.
The government literally pays them and promotes them as a symbol of the nation - how is that not offering them "wealth, opportunities for their children, or a popular reputation"?
They have a tolerance of how much resource can be wasted. President's kickback isn't worth anything when the resource wasted can actually cause problem for the nation(Only at this point the monarch will start paying attention). So i am pretty sure they both are similar in this context.
This is categorically different from a democracy where the leadership's wealth is separated from the nation's. If the laborers die, the President is not impacted in the least. A President may choose to spend your life and money to take over new territory and increase their power, but they are spending your wealth to do it, not their own. If the resources are mismanaged you are harmed, not the President.
Next this doesn't even work out if we replace the monarch with an automaton which faithfully executes economic theory.
A monarch of 100M people might spend 10,000 people for some end that on net increases his wealth whereas a president of 100M might find the rest of the people so appalled at the 10k corpses that they vote him out of office.
If you look at how monarchs have actually and in fact spent the blood of their people historically we wouldn't have to argue theory. Throughout history monarchies have spent their people liberally in proportion to their ability to do so. You can look at recent history at a bunch of nations that are still technically monarchies you will note that their period of moderation directly coincides with the waning power of the very actor you claim somehow serves to moderate the state.
This understanding is completely outside of actual reality. The statement "Monarchy is superior to democracy in a very important way" goes on to explain why you believe the monarch has a greater incentive to avoid wasting the lives of his citizens. This isn't even close to true. A monarch who loses 1% of his "wealth" in the form of dead people still has the other 99%. A President can lose his job, his wealth, and his personal power for less.
A president is incentivized not to put upon the people more than they can bear with the understanding that all they have to do to be rid of him is check a different box next go round. A monarch faces the same risk but only if the anguish of the people is enough for them to spend copious amounts of their own blood in violent revolution.
This difference in resistance is the dominant difference in motivation NOT a proprietary feeling towards a kings subjects. This is well born out by an ocean of blood spilled foolishly by monarchies through history. I don't misunderstand you. I don't think you have properly understood history.
Also, the Monarchy is effectively a mechanism for it's powers to be controlled by ministers, this means parliament can and has been effectively shut down temporarily to avoid debate at the whim of Government Ministers against the will of parliament itself.
The monarchy in the UK only seems to benefit themselves and tabloid newspapers. While the Queen has been a dutiful monarch (whos work ethic I happen to respect), the rest of her family seem pretty terrible with their crazy shenanigans.
What work does the Queen actually do?
As an American citizen it is why I don’t mind presidents that I don’t like have a large secret service revenue, fat salary, and so forth both during and after their presidency.
She's like a grandma the nation respects, even though you suspect there might be a sinister side to her.
Also, the wider principle is that there shouldn't be an untouchable hereditary ruling class at all. It's an absolute nonsense.
One of the key problems in many human societies is that most of the wealth and power ends up concentrated in the hands of the undeserving few, who use it to further enrich themselves and their families, at the expense of everyone else.
Doesn't that just shift corruption to the next party? I don't see why logically switching the ruling party would abolish corruption. It might even increase it.