All the top of head no particular order
-) Being able to survive disease would be spun as more evidence of the nobility's superior nature and right to rule. This would delay or even block the evolution of democracy.
-) Nobles might use the offer of magical cures as a means of controlling the upper tier of the population. "Behave or when your kid gets sick, I won't help."
-) If the world has a universal religion like Buddhism, Christianity, etc., then the hogging of cures might causes serious moral revulsion among the majority of the population. "Death is the great leveler," and the idea that one's taxes went to pay for keeping the King's children alive but not your own might create political blowback.
-) A mere taboo will not stop people from trying to heal their children. I can infer from you adding this to the list that you don't have children. I personally would do horrible things to almost anyone in any numbers if doing so would save the lives of my descendants. I wouldn't give a bleep about custom, tradition law, fairness, or the consequences to myself. All parents feel this way when it comes down to life or death for their children.
Steven Spielberg said of the ending of Close Encounters of the Third Kind, where Richard Dreyfuss's character left his wife and children to fly off with aliens, that it seemed credible when he was single and childless but, after he had his own kids, the character's action seemed wholly implausible.
So, if you want to prevent children from being treated, then you'll have to create an actual magical barrier that no one can get around otherwise your readers with children will be yanked out of the story.
-) Greater political stability:
-) Assured succession: In the medieval world, kings functioned like the keystone in an arch, with the most powerful nobles being the stones pressing in from the side. The king had to be militarily powerful enough to bring the greatest nobles (e.g. dukes) to heel if they started private wars, refused to obey the law, or made foreign alliances. Moreover, kings were expected to be warriors who led from the front.
Basically, the medieval world was run like the mob and the king was the godfather or, more accurately, the mob is run like medieval kingdom. In either case, the guy at the top must have troops, military skill, and will to bust the heads of the most powerful when needed.
By the medieval age in Europe, the Germanic tradition of the legitimacy of a king being conferred by election had evolved to right by inheritance. If a king died before while his crown prince was still a child or, worse, died without heir, the kingdom lost its keystone and began to fall into internal war and disruption.
Improved medical care for royalty would provide greater stability by (1) increasing the lifespan of a monarch such that their heirs would be grown adults by the time the monarch died (this would also make policy more consistent and stable) and (2) would allow more heirs to survive, providing a greater talent pool from which to select an heir.
The best example of this phenomena would be the difference between Henry V and his son Henry VI. Henry V was the most effective political and military leader of his generation, accomplishing the feat of providing year-round logistics so his armies could campaign year round instead just through the summer months. It was that, more than his use of longbowmen, that allowed him to fight his way to the throne of France.
But he only reigned for the seven years 1415-1422, dying of dysentery while on campaign and leaving only his infant son as king. At first, the great magnates of the kingdom cooperated to keep things running smoothly and, for a decade, Henry V's brother was on hand over in France to slap nobles down when needed but, gradually, the English government became more dysfunctional.
Things got worse when Henry VI actually assumed the throne. Raised in a nightmare environment where he was constantly pulled this way and that by the most powerful men in the land, by his teens he was passive and likely to agree with the last person he talked to. He was a soft, unenergetic person with virtually no physical skills at all, much less a warrior. His inability to serve as as a keystone and contain the growing feud between the Duke of Somerset and the Duke of York triggered the War of the Roses.
Likewise, Edward IV managed to bring stability back to England but he died of natural causes, likely aggravated by his obesity and gluttonous living, leaving behind Edward V, a twelve-year-old. Edward V ended up murdered by Richard III by most accounts, triggering yet more instability which eventually led to Henry Tudor becoming Henry&bsp;VII.
If Henry V had been cured of his dysentery and survived another 20 or 30 years, his brilliance might have cemented an empire of England and France. Then either Henry VI would have assumed the thrown at middle-age, having had a less insane childhood or, if he still simply lacked the right temperament, could have become a monk (which he really wanted) and one of Henry V's other sons could have taken his place.
-) More first sons, raised to be kings would assume the thrown: In English history, there are two crown princes, both named Arthur, who died of disease thereby altering history.
The first was the son of Henry VII, who died shortly after marrying Catherine of Aragon, which triggered a whole avalanche of consequences when his younger brother, crowned Henry VIII, took the throne and Catherine as his wife.
The second Arthur was the first son of James II, the elder brother Charles II and his polar opposite in all regards. Charles was short, bowlegged from rickets, had a stutter, and had be raised in Scotland under the tutelage of radical Presbyterians whom he hated all his life. He was a Francophile with a Catholic wife who sought to become an absolute monarch. Arthur was raised largely in England, was an active Francophobe who probably would not have married a Catholic, and idolized the idea of a constitutional monarch. While Charles loathed such free-spirited men as Sir Francis Drake and other sea adventures, Arthur worshipped them. Had Arthur become king, it's unlikely the English Civil War would have occurred and that the transition of England to a stable constitutional monarchy might have occurred without violence.
Of course, then there would have never been the Dutch invasion under William III and, with it, the importation of Dutch innovations like insurance and banking, but it's likely enough that would have occurred by diffusion anyway.
-) More surviving children means more alliances through political marriages. In principle this would create more stability but, with more marriages and alliances, the value of each individual marriage alliance would decrease. You might see an effect common in medieval Islam in which a Sultan, Caliph, etc. had four wives, all owing to alliances. Each alliance was weaker than those formed in the monogamous culture in Christendom.
-) No mad kings: Henry V conquered France in large part because his opposite (IIRC, Charles II) had intermittent attacks of severe mental illness. A monarchial government can deal with a dead king, a child king, and an inept king to a degree, but one that goes utterly nuts is a nightmare as no legal means exist to remove him for incapacity. Henry VI lapsed into some sort of catatonia, a waking coma, for 18 months after learning of the fall of English France. England went to pieces during his incapacitation because nobody really had the authority to rule in his place.
-)Too many heirs not enough land: With larger royal families, that would mean finding titles and lands for more sons. It would also mean more opportunity for jealousies to lead to arm conflict. A disturbing number of medieval nobles got whacked by their own first- or second-order kin.
Even worse, as the generations progressed, more and more people would have royal blood and thus some claim on the throne. If the immediate heirs of crown did all get killed by a sudden combination of accident and warfare, then the resulting scramble for the throne would be a dogfight.
It might lead to more expansionism, more crusading/jihad, or more exploration.
-) Possibly more inbreeding: Historical European nobility are so inbred they make the proverbial (and largely fictional) inbred hillbillies look like paragons of genetic diversity. With larger royal families, all intermarrying, the possibility of marrying too closely over successive generations increases. (Again, this was and is a problem in high status Islam. The wealthy produce a large number of half-siblings, who marry into similar families, producing another batch of half-siblings, and, since marrying cousins is still legal, genetic consequences can be severe.)
On the other hand, if you can fix those problem with magic, then maybe not.
-) Mages who could heal would become viewed as assets to be captured and controlled. Given the intense need for lineage and dynasty at all levels of society, a king is not going to bargain excessively with a mage or take "no" for an answer. He's going to say, "Cure my child and I'll let you live in a fancy estate as your prison. Don't, I'll torture and kill your entire family, all your friends and your little dog too." And he'll keep doing that until someone capitulates. Once he has such a mage, he won't let him out of his control and will probably keep him nearby.
It's likely that an institution would develop in which the mages would be considered automatic retainers or even slaves of the monarch with the innate duty to see to the royal family's health for the good of the realm. Those that didn't would be seen as traitors threatening not only the lives of the royal family but the peace and prosperity of the entire kingdom.