This is less of a historical question and more of a conceptual question. I'll take a crack at answering it. Note however there are some problems with the question. Feudalism has a wide range of possible definitions. Many has more to do with a relationship built on ritual, power, and loyalty, rather than bureaucracy. So in many ways this question, and answer, isn't really about feudalism.

With that out of the way, let's tackle the issue. While the math in and of itself is not incorrect, (or rather because we're working with hypotheticals there's no way for it to be wrong) there are many things you have failed to consider in your assumption.

1) The kingdom quite probably did not have the resources to hire the 15 administrators to begin with. A kingdom must start from somewhere. Before the modern day, very few armies were kept under permanent pay of the state. More often, a system existed where service was an obligation. Armies were called up for the campaign, and they would be paid for the campaign only (if even that, often they were not paid at all), and in the end they go home. Let's say you lived in such a society. You raise an army to conquer a new province with an army of men who are obligated to serve under you, supplemented by mercenaries. You only have enough money to pay for the campaign, as under such a system you neither have the ability to raise large revenues nor the need for such. And if you had more money than needed for the campaign, you'd probably raise a larger army to increase your probability of winning. So you've won. Your men, now not expecting any further payment or loot with the fighting over, want to go home, for their farm or shop or estate is what keeps them alive, and without them home the farm/shop/estate could very well go under. The same goes for your commanders and bureaucrats. They need to go back to administrating their own provinces. They are busy enough with their own realms and have neither the spare time nor incentive to help you rule your new provinces for free. And, having spent your treasury to win the provinces in the first place, you don't have the money to pay for years of administration. Of course, once the administration of your new provinces is set up in a few years they'll pay for themselves, but how do you get to that point in the first place if you don't have spare resources and manpower to get there? The only solution then is to offer the resource potential of these new provinces, like future taxes, to people to entice them to run the provinces. Whether it's to the mercenary leaders, your lords, or leaders native to the provinces who had surrendered and sworn loyalty to you. And you have little choice, for running a province is expensive, these men would need that income to run the province to begin with, not to mention no one wants to just break-even and they want actual rewards for their service. So because you don't have the money to hire all the necessary administrators for your five new provinces on top of everything else you need money for, your only choice is to appoint a lord to each one, and give him the income from that province, to have him figure things out himself.

2) Your kingdom is in a protracted civil war and you need the loyalty of your generals. While we started at a decentralized administration above, we don't even need to in this case. Many years ago civil war broke out. The reason was often due to unclear succession but could be anything. It turns out the sides are fairly well-matched, and the war drags on. Let's say both sides have five generals commanding five armies in the front line. As the war drags on the loyalty of some generals begin to waver. They could be ambitious, or angry they got passed over for promotion, or their army is sick and tired of fighting a war that's not ending against men they see as their brethren. Maybe at the start of the war they loyally served your father, but he has since passed away and they frankly don't feel the same loyalty to you, and the invitation to switch sides from the cousin on the other side is looking more and more enticing. So a general writes you a letter demanding you increase their pay by 20%. Now he knows your treasury is running empty from prolonged warfare, so he tells you he'll just keep the income from the province where he's been deployed to for all these years to pay for this, and send nothing to you. And he's angry you have punished one of his lieutenants last year for pillaging in the province and demands the judiciary power in the province as well. Currently it's 5 on 5, and if he switch sides it'd become 4 on 6 which might very well cause you to loose. And let's be honest, your opponent probably gave him a similar offer. So do you say "no" in an attempt to keep your bureaucracy centralized just to then loose that bureaucracy to your opponent? Of course not. You say "okay, fine." All the other generals learn of this, and even the loyal ones politely request you for the same thing as he just want to use the money more efficiently to increase the size of his army with more local militias to help you win the war. And all of a sudden your kingdom has 10 semi-autonomous provinces. You end up winning the war, probably by successfully enticing some of the enemy generals to join your side. And you ask them to start sending their income to you again and return their judiciary powers. They politely laugh in your face and remind you that you are broke from all the years of warfare, and they have veteran armies used to looting and income from their own provinces so they don't need you. Do you take a risk and start another civil war, which may very well force you to make the same concessions or more, or do you sigh and allow things to continue the way they are?

3) The five provinces really do not want to be ruled by you. The time is before national public education, before railroads, before television or radio. Or those could exist it doesn't really matter. The men of these provinces really do not like you. You could be a recent conqueror or not. You are hundreds of kilometers away. Your spoken language, perhaps written language as well, is quite different from theirs. Your laws, customs, and religion do not align with theirs. Ignoring any personal grudges, they just feel that you do not understand them and the issues they face and, quite frankly, you do not. The same problems apply to all the men around you that you might send to their lands far far away. They are so angry that there are talks about rebellion. You might want very much to crush them under your boots and force their obedience, and under usual circumstances you may very well be able to. But what if at the moment you can not, perhaps because all your armies are busy fighting other wars. You might even need their armies, even while many of the soldiers from these provinces rightly share their countrymen's annoyance at the way you rule them. Even if currently the country is at peace and the treasury full, a rebellion will likely cost you many times the money you can get from the taxes from these provinces in order to crush it, and if the rebellion drags on your neighbors may see an opening for invasion to settle some old grudges. One of the province's governor write to you that your external foe has been trying to fan the discontent in the province and entice them to rebel. However he does see the some advantages of staying loyal. He just wants some concessions so that he can convince his countrymen to stay loyal. Who knows, it might even be a thinly-veiled threat. Nonetheless, the concession he wants is for the men of their province to choose their own bailiff, reeve, and steward, people who they know personally and feel understands the issues facing them better than you do. A larger portion (or all) of the provinces' tax would be used to to administer their own province, so won't be sent to you. They also want to have their own laws and keep their own customs and religion. They basically want autonomy. So, do you give them what they ask so they don't rebel, or do you keep your centralized system and risk their rebellion?

Many of the rulers in history were very aware of limits of loyalty and control decentralized rule causes. When given the chance, many do try to switch to a system of centralized administration, appoint the bureaucrats from the center, conduct regular census, standardize the law, language, religion, etc. However the existing structure of power often prevents the central ruler from amassing enough resource to achieve a decisive transition to centralized administration in the first place, for few rulers central or local want to give up their own power to someone else. I've mentioned above many situations we see in history of why a decentralized administration come to be. Attempts to centralize rule from this situation face much of the same challenges as outlined above. Centralizing an administration might hold great rewards, but a failed attempt could bore great costs. If you decide to replace five hereditary lords in five of twenty provinces in your country with personally-appointed stewards, the lords may very well rebel and support your brothers' claim to the throne.