Monarchy is a rather broad term so Im going to explain why I believe the constitutional monarchy of the UK is by far the best political system.
A bicameral legislature is widely employed globally and is demonstrably the ideal democratic legislature. A discussion about bicameral legislature is a different topic so Im going to speak on the premise that it is the accepted legislature.
Every legislature needs a safe guard and for most bicameral systems this is achieved via a president or head of state. The advantage of a bicameral legislature with a head of state is the division of powers - for example the army and the courts in the UK pledge allegiance to the head of state rather than the legislature. This provides stability and a transfer of powers to the subjects of the constitution.
Furthermore some legislatures have written constitutions however I believe this is highly flawed - written constitutions can become needlessly partisan and the best example of this is in the US which is slow to evolve and adapt and feeds into the partisanship of US politics. The US constitution has almost become a religion and is stuck, slow to keep up with its peers and now seems rather backwards to other developed nations. An unwritten constitution defended by a monarch on the basis of trust provides a higher degree of protections.
The constitution - a body of fundamental principles or established precedents - should be thought of as the basic acceptable rights of the people - and that is always changing and evolving and as such should not be written in stone. The advent of the computer and various other technologies such as future AI is a testament to that. A head of state (president) is essentially the defender of the constitution and needs to adapt to the ever changing challenges faced by the subjects.
In republics their method of election varies - it can be a plebiscite or can be elected by a council. I argue that this is inherently flawed. My main issue with elected head of states / presidents is that they are more easily corruptible than monarchs (herein I will refer to a monarch as the head of state). A president's/monarch's role is to foremost defend the constitution - which as Ive said should be unwritten - and the constitution is essentially for the betterment of the people. The monarch can be considered as the defender of the people's interests. If the monarch fails to defend the interests of the people then they shall fall out of favor. The reason why I believe a President - elected via council or plebiscite - is more corruptible is simple: stakeholders. A constitutional monarch is likely to own swathes of wealth and property in a country - I propose this is essential requirement for a monarch. Theirs and the prosperity of their heirs depends on their success in carrying out a relatively simple task - defend the constitution. Failure to do so not only loses them their source of income - but also the income of their heirs, along with their vast wealth and possessions. Alternatively, a president who, for example, was a successful politician or financier for the past 30-40 years prior to their election as president (e.g. Macron), is not phased by such a predicament - they have no invested wealth and can return to their previous profession if necessary.
Ive been drinking a lot tonight so this might sound rather verbose but we talk the best shite when were drunk dont we?