This way madness lies.
I'm working on a hypothetical political system similar to a monarchy except that the (conscripted) king in question is essentially a slave to the nation,
The word "democracy" leaps to mind. :-)
a very well cared for slave but they have no personal freedom.
Kurosawa's Kagemusha sounds like this.
The idea behind this is to prevent corruption
by ensuring the king has no personal bias, no conflicts of interest, that he (or she, gender isn't relevant) cannot be bribed or extorted.
A normal all-powerful monarch would have no conflict of interest (just their own interests), could not be bribed (they own everything already) or extorted (try blackmailing someone who can have you killed very slowly with a nod).
You also seem to think that being corrupted or bribed comes only from within the person. This is quite wrong: pressure and manipulation from other people will corrupt someone - it's the basis of all politics.
Personal bias cannot be eliminated. You would need to eliminate the person. Over time, no matter how neutral they start out, they would inevitably develop a bias in some way or other.
This is achieved by preventing the king from having interpersonal relationships of any kind (no friends, family or lovers), ensuring all of his/her needs are completely catered for and that he/she is almost completely isolated from society's influence.
What you are describing is a type of dehumanization.
Humans have been placed in prisons like that before - they go mad. It's not possible for a social animal like a normal human to survive without some (or many) mental problems caused by isolation. When prisons want to punish prisoners experience has taught us that completely isolating them will have a terrible effect on any but the most unusual (and not necessarily normal) people.
For sane people all of his or her needs includes companionship and warmth which you're excluding. While many people can manage celibacy, the idea of no friends or family is one that would result in a psychological disaster. The worst prisons have not been the ones where physical abuse is practiced, but the complete isolation from others - it is a torture that robs the soul.
Orphaned infants are raised separately by highly trained carers wearing full body covering form-concealing clothing, scent concealing perfume, full-face masks, and they all move in the same highly ritualized way. These carers are on a rotating roster so even if a child managed to identify one they would see that one so seldom that they wouldn't be able to form any personal attachment. Instead, they're encouraged to interact with all carers as though they were the same person, specifically a characterized mother figure that represents the nation itself raising and caring for them.
The worst aspect of this (from your point of view, although not from that of the poor sod who has to suffer it) is that your individual will develop no sense of empathy with other people.
Empathy is not something you get without exposure to seeing how your action affect others (and in particular how your cruelty can cause them suffering).
So your ideal ruler will no interest or empathy for his country or the people in it.
They would simply rule by decree in an isolated and abstract world which they do not understand.
As the orphans grow up they're schooled with a particular focus on history, philosophy and political theory. When they come of age the current king is disposed of (kings are replaced every 20yrs or so) and the orphan who is deemed to have the best disposition and aptitude for the role is ordained to be the new king, after a year or two without the new king showing instability the other candidates are deemed unnecessary and disposed of.
Wow, that's so naive.
Real world :
Either the monarch spots that his position is not guaranteed and they're not safe (what does "disposed of" mean ??) or someone else in court who is crafty decides they're going to mount a coup d'etat and control the puppet monarch themselves. As you've taught the monarch all this political science stuff they're going to be well grounded in how conspiracies work and how to get stuff done.
So in short order, this monarch, one way or another, will gather a solid and entirely loyal group of personal attendants (the expression "Praetorian Guard" comes to mind). Once everyone close by is controlled by fear, money or apathy (and apathy is a real concern as in your system it effectively makes no difference to them who is in charge), then the expert politician will rapidly move to take over all power.
And in the process say goodbye to their days as a lonely, loverless, friendless prisoner. And flay alive the b*rst*rds who made the monarch suffer that in the first place.
What your process will create, in fact, is a ruthless individual skilled in manipulating others and political machinations with a huge self-interest in changing things to suit their own needs and no interest or empathy at all in the lives of others.
You've made Stalin, or Hitler or maybe Genghis Khan. Congratulations.