"And a more cost effective head of state would result in a greater profit."
True, as would having a more 'cost effective' anything - I assume you mean "cheaper"
We could gor example mimic the French, who spend significantly more on theirs.
Or if you'd rather look at Europeans monarchs, and do it relative to population of that country, only Spain in cheaper (Norway is most expensive, followed by Holland, Denmark, Sweden, Belgium, then UK, then Spain) Source, analysis from that notorious pro-monarch newspaper The Independent. Liechtenstein and Monaco royals don't feature in the cost per taxpayer list, but were noted in a separate list as being wealthier as royal families than the British.
So I think it's wrong to see monarchy as being inherently more expensive than a president, or dictator-for-life, or whatever other arrangement someone comes up with, because it isn't. Nor to see the British monarchy as expensive among monarchies, as it isn't.