You're 100% right. The UK would be served just as well by a nonpartisan president.
Stay Classy, Britain.
Oxymoron.
You have to vote for a president right otherwise you end up with ... well Oliver Cromwell. A Dictator by any other name.
Nonpartisan in the sense that the Queen is nonpartisan. So of course, partisan. Still, nonpartisan like not publicly wading into political debates, and not opening up the office to the vulgarities of electoral politics.
For example, Germany elects its president by convening both the Federal and State legislatures together, who then vote. By pulling in all the regional parties and legislators, the President has to have broad support across party lines. It's also convention that the President doesn't wade into daily political issues. The result is a largely ceremonial office, usually held by a broadly respected statesperson with former government experience. Which is what you want for the "emergency button" you hope you don't have to push in the first place.
This is relevant because Western monarchs claim the office of Rex/Regina.
Unelected officials should never have significant power. This led to a harmful coup d'etat in Australia. It's not a ceremonial office, and simply the fact that the emergency button exists makes it possible for perversions to continue and cause serious issues (and it has).
Partisanship is very far from the only cause of dysfunction in governments. I'd argue it's not even the worst.
However, even if it’s generally accepted that the child or close relative of the current leader should become the new leader there will eventually be back and forth over time between the royal family and other members of government. In theory they might have absolute authority, but in both practice and tradition there will be many informal limits.
In general, monarchies pretty much have had the absolute maximum of power concentration possible without disastrous consequences.
Where presidents have actual executive power such as USA, France, Russia, the issue you bring up can, at least in theory, occur.
I'm not sure if the postal service still is, and the prisons could probably be let go - but I think there's a lot of land, especially in the south owned by the crown.
The royal family does have many billions in personal holdings as well, of course.
The Crown and Queen Elizabeth II are different persons. Queen Elizabeth II is a natural person. The Crown is a legal person, a corporation sole – a corporation with only one member, who right now is Queen Elizabeth II. Actually the Crown is not a single corporation, but a number of legally distinct corporations which all have identical membership. There is "the Crown in Right of the United Kingdom", "the Crown in Right of Canada", "the Crown in Right of the Commonwealth of Australia", "the Crown in Right of New Zealand", etc. All up, there are 16 nations with Queen Elizabeth II as monarch, and each of those nations is a distinct corporation "the Crown in Right of X". But, also there exist distinct "the Crown in Right of X" corporations for each Australian state (6) and each Canadian province (10), and also each of the three British crown dependencies (Isle of Mann, Jersey and Guernsey). So there are 35 legal persons (corporations sole) "the Crown in Right of X", and one natural person who is presently the sole common member of all of them. Each of those 35 corporations has distinct assets, which in turn are distinct from the personal assets of the natural person called Elizabeth. If any of those 35 entities became a republic, the assets of the "Crown in Right of X" corporation would pass to its republican successor.
That said, there are some Duchies that are privately owned. In my opinion, these should be seized by the state during the establishment of a republic - if we're going to abolish the monarchy, we need to go the whole way and strip each member of all their inherited wealth too.
It's the very least they deserve, and would provide a much needed clean slate for New Britain.