May I suggest reading "The Night of Madness" by Lawrence Watt-Evans? It is a fantastic exploration of the whys and wherefores of Wizards (and other magic users) guilds and what functions they serve internally and what functions they refuse to serve externally. (As well as being a good book.)
The point being that the Royals would do that, except that:
A) Are the wizards going to put up with it? While a Wizard would not do well to sneer at a King, especially in 5e a group of Wizards is really NOT something a King wants mad at him either.
B) Succession. Princes like becoming Kings before they by of old age, at some point they become highly motivated to move the succession problem along.
C) Interrealm politics. While dynastic kingdoms tend to think in the long term in a way almost impossible for modern americans to grasp (an effect that would only be hightened by the presence of long-lived races) they are still going to be non-plussed by the presence of neighbors who can actaully see 200 year plans through to fruition. So any kingdom that has access to a transmuter willing to sell longevity had either better share him or be willing to face all their neighbors on the battlefield and/or survive a series of assasination attempts.
OTOH We already have both extremely long-lived races like elves, as well a host of practical immortality options like undeath. There is a lot of good potential to explore and exploit the effect of turnover rates amoungst the royals. Lich kings, vampire counts, Elven royals, Dragons, heck even full fledged God-Kings.
Might be a fun campaign to run something where the world has had a long period of stable rule under a handful of near-immortals but now a change is coming and the immortals are too set in their ways to shepherd the world through what is to come successfully. So it's up the our heros to ... move along the process of changing leadership. Or at least, that's what the Hermit says...
