Moreover an elected head of state (in terms of a presidency) would undermine the (very successful) Westminster model of democracy. This relies on the sovereignty of parliament and the consensus that parliament is the only body with the legitimacy required to govern. An elected head of state could clearly claim legitimacy from national constituency and could leverage this legitimacy into political influence. This is not a good thing.
Um. I take it you're not familiar with Ireland or India then? Both systems developed directly based on the Westminster model. In both cases the president does essentially what the Queen or Governor-General does. (Australia, for the record, has something often called a "Washminster" system - because somewhat unusually among Westminister systems we have a strong fully elected Senate which is the equal of the Lower House and can block bills)
This would seem to suggest you believe that parliamentary-supremacist republics aren't a thing. Nothing about having an elected head of states suggests they have to wield power or be anything other than ceremonial, and there are literally dozens of examples of republics where this is shown. In fact, outside the Americas, only a minority of republics have a strong active head of state. Can you name the president of Germany, of Israel, of India? Yeah, exactly.
Finally, the international benefits of the monarchy are often ignored. The monarchy is what unites the commonwealth and thus creates and excellent network of diplomatic links. For Britain, the monarch constitutes and excellent international ambassador which greatly helps the country maintain close relationships with its former empire. An elected head of state would make this whole system unstable.
This is actually just factually wrong. Firstly the Commonwealth is essentially pointless, and secondly, only 16 of the 54 members have Elizabeth as their head of state. Five others have autochtinous monarchies, but the majority of Commonwealth members are in fact republics.
None of your post says anything about the compatibility of monarchy with liberal democracy, though. All it's actually saying is that "there are benefits to having a ceremonial monarchy" (or rather, for ceremonial
heads of state), which in fact concedes that to have a liberal democracy, you must neuter the monarchy to the point of ceremonialness, so the
mon doesn't actually
arch.
I should say I do actually accept and support the idea of a ceremonial head of state. I'm not convinced the presidential republic model practiced in the Americas works very well. Sometimes I think the American President is treated like some sort of elected god-king.