I think the issue is Corlys is prideful enough to want his house's name as the primary and his dragon riding family members (aka the people we want on our side the most) follow in lock step thanks to patriarchy.
Okay, it's just that by broad category this is the kind of issue that
CAN be resolved, as opposed to the kind of issue that would seem nearly impossible to resolve short of open warfare because truly irreconciliable interests are at stake.
I like this idea.
A fair amount of Fanfics involving the Iron Throne (uplifts specifically) tend to involve expanding the Kingsguard in several different ways (Squires, Sworn Men At Arms, making them 7 by 7, a whole ass thousand strong Septguard) but this is the first I've heard of arranging for an institutional "retirement" process.
I can see a lot of potential to expanding the very concept of the Kingsguard into something more, though we would have to be very careful to preserve the sanctity of the name itself.
Eh.
The Kingsguard functions
specifically because it is small. There is a fairly limited supply of highly skilled knights in all of Westeros who are willing to swear to celibacy (possibly chastity) and forgo actually getting to rule their family estates in exchange for being one of the Kingsguard. To expand the Kingsguard significantly we would need to do one of two things.
One possibility is to increase the incentives... which is hard to do, because we can't actually "pay" them in greater honor and prestige, and since the entire
point is that the knights forswear most of the things that ordinary Westerosi fighting men fight for.
The other is to accept a decline in quality. In which case I suspect we'd be getting, not a larger force of Barristan Selmies and Arthur Daynes, and probably not even a larger force of Jaime Lannisters and Mandon Moores, but rather a larger force of Boros Blounts and Meryn Trants.
...
Furthermore, the Kingsguard frankly does not
need expansion. Its purpose is to serve as close personal protection for the actual monarch and their immediate family. SInce this is an era where things like bolt-action sniper rifles do not exist and so there is much less need for a small army to maintain a 500-meter-radius security perimeter around a public figure if you want to keep them alive reliably.
Making the Kingsguard dramatically larger also increases the vulnerability to plots, because it's hard for any one person to keep close track of the personal loyalties of that many other people. Since by definition the Kingsguard are among the very small number of people actually in a position where their treachery could be
really devastating, that's a big deal.
The point of institutional Kingsguard retirement with honors is not to cause some kind of radical expansion of the Kingsguard, or even to lay groundwork for it, and it's certainly not intended to increase the institutional capacity of the state (I'd create a new institution for that, or work on the goldcloaks, or
something).
The Kingsguard is a knightly order with a highly specific purpose, and the only point here is to make sure that there actually are, at all times, seven
combat-ready knights in the order. And not, say, five combat-ready knights, one man who can stand around in his armor looking vigilant but probably wouldn't win a serious fight against credible opposition, and one outright cripple.
The combination of Prestige and Royal Authority helps provide additional separation and potential loyalty compared to a standard employment contract while also allowing us to set certain enforceable standards on quality and training regimes.
The Kingsguard's prestige comes in large part from the fact that it is extremely selective and that you have to make real sacrifices to join. Expanding the force or loosening the requirements would dilute the prestige.
Or in other words, it gives us an angle to form a core to build an army around, or at least Special Forces equivalents as Dragons are still in play, during a time when anyone having anything resembling a Formal Army would be seen as a vast over reach of authority and intent.
The Targaryens absolutely could raise an army and effectively do; it is the routine prerogative of any of the Lords Paramount to have their own army. The catch is being able to afford a large, uniformly equipped body of troops, because even with the Targaryens' wealth we're never going to be able to pay to equip as many armed men as Westeros as a whole, or even just the Crownlands, could theoretically muster. Well, not without the work of generations going into building state capacity.
But no, we absolutely can just
have an army, it's fine.
It's just that as a practical matter, if we want to sustain an army of more than, oh... 5-10 thousand soldiers, we're probably going to have to start leaning on vassals who can get proportionately more taxes out of their lands than we can, and call up those banners when it's time to go, which is how feudalism normally works.
Edit: Also, preferably the Royal Family will actually become large enough that things like "cousins" and "second cousins" exist, and having a larger Royal Guard would help with future proofing.
Such a large royal family does not require constant elite bodyguarding, because while someone killing the king's second cousin is
bad, it is not an immediate political crisis that threatens to throw the realm into civil war. The people who actually need bodyguarding by elite knights are the monarch, their spouse, and their children, most especially the heir.
The Kingsguard is not the United States' Secret Service, nor is it the United States' special forces units, nor is it the United States Army, nor anything else like that. It is a very small order of knights who swear very strict vows for a very specific purpose.