Great posts Scorox!
I like a lot of the ideas and they'd be great solution, but I think the problem analysis is incomplete/the game's problems are worse.
You write that "It is an incredibly cleaver idea to introduce the 3 different roles (Gov, Duke, and King) in a systematic fashion to accommodate for players different levels of activity and wish for responsibility in the kingdom."
The problem is that I don't think that feature was actually meant that way. I think the progression from governor to king was considered to be a purely temporal, from newbie to veteran player.
Governors would be a mostly supportive role for the kings who actually compete to win the game, while they in turn are less threatened and can learn the ropes and participate in the game for a world or two without actually competing much.
I think it was inteded that over time, *every* veteran player would become king - activity or wishes not being a factor.
That's why there actually is a restriction - not everyone can select king role, but players with a certain minimum prestige. The original design expected players to want to play king so badly that they'd need to restrict access to the role.
Still, many of the kings in-game are zero-prestige newbies in their first round, because of all the veteran players this game has, way too few choose playing king voluntarity. That would mean, since without kings -> no kingdoms and without kingdoms -> noone collecting victory points to compete over the game (no kings meaning to the game as it is designed that just noone wants to play competitively), the game would effectively become Sim City. That's why the game dynamically allows even newbies to become kings, since with not enough kings, the game opens the "floodgates" and forgets any requirements and allows anyone to become kings, just so there are at least some player playing competitively.
And that's where the root of the problem starts:
Yes, quality of kings is low.
That's because the quality checks are ineffective.
The quality checks are shut off in first place because if they were effective, we'd have noone playing king at all.
And that's why having stricter checks doesn't solve the actual problem - you'd just end up without kingdoms, because too few people like to play kings, and those who do want are bad, so with extra hard checks, even those who like to play king right now wouldn't make it anymore -> no kings remaining.
The problem isn't "how can we better weed out the suited candidates from the bad ones from our huge pool of candidates for the king role", the problem is "how do we get enough candidates for king in the pool to allow us to be picky in first place."
You see, if you consider that this may have been the original intent of the design, the obvious remedy to all the players' woes is to "just pick king as your role!":
Players complain that they prefer much smaller kingdoms - but to get smaller kingdoms you'd need many more players playing kings, and to immediately get many small kingdoms all the community would have to do is to just pick the king role damnit.
Many players complain that their kings are so bad and that ruins the game, and that governors are really at such a disadvantage, and that they can't change anything about that as a governor, but that they still want to play competitively and win, and they don't pick the obvious remedy the game offers for their plight: Playing as a king themselves. So play king and you can change it!
I just think the original design vastly underestimated how much different the appeal of "manage a small group, be responsible for everything, and get blamed for everything" is to players (original idea assuming that everyone would want to do it) compared to "having no responsibility whatsoever but still be on a winning team", which is essentially the governor and duke roles.
To solve the underlying problem, we need to somehow fix the attractivity of these roles, so that we have more candidates for the king role, and then we can get picky.
But the game not-being-picky-enough is unfortunately not the root cause of the game's problems, IMO. That'd be really easy to fix, as you describe. It's harder to fget comptetent leaders in a community where noone wants to play leader. That's more of a complete failure of the original idea of seperate roles in itself, and more of a hint that the game doesn't attract a diverse enough playerbase to fill 3 distinct roles in first place.