Rpg sessions are ideally greater than the sum of their parts. One of the reasons that posters end up talking past each other is they can't see context in the narrow bandwidth that forum posts allow. Different campaigns aim for different levels of directness, action, plottedness, PC importance, rules importance, grittiness etc. etc. These differences mean that situations that on paper are the same work out very differently in different games.
For instance, the status of adventurers varies from DM to DM. In my game adventurers are anomalies in society even at 1st level - they aren't peasants, most aren't nobles, they may be technically commoners, but they are heavily armed and free to travel and make decisions for themselves. They are a potential threat to the rulers, but also useful as expendable troubleshooters to deal with problems like goblin raiders. The typical campaign world isn't as socially stratified as the real medieval world was. And most of the rulers in my worlds have class levels, as it makes it easier to justify them having power in D&D worlds where class levels grant raw power.
In a fast-paced action game, guidelines common in TV shows like "the obvious solution never works" can be inappropriate. It's bad for games like this to stall, which can happen when there's no obvious way to the players to solve a problem and it has to be solved for the plot to proceed. It's better to avoid a situation like this in the first place, but once in it, DM force is one way to solve it, whether it's "ninjas attack" or help from a mysterious stranger, just to keep the action moving.
Now, my take on the Chamberlain dilemma (the poor guy is horribly overworked, I know, I hope he gets overtime). The choices on how to evaluate this in a particular campaign are affected by many, many factors, too many to go into comprehensively in a short post.
How I would evaluate it in one of my campaigns depends primarily on the level of the PCs at the time. I generally run a conventionally heroic campaign so the PCs will become local heroes in the early levels. Normally they are offered petty nobility by mid levels. Low level adventurers in my games typically won't need to meet kings for plot reasons, and aren't important enough to get special treatment, barring special PC backgrounds. I will cheerfully tell the players this OOC if they come up with the idea themselves - if they fixate on it I may work out something to keep everyone content.
At mid levels they may want or need to see the king for something, and should have sufficient reputation, resources, favours earned and allies to have some chance to do so if they haven't become outlaws or something. It's the chamberlain's job to stop time wasters from seeing the king, so the PCs appearance, reputation and knowledge of protocol will have a bearing on their success. If the chamberlain doesn't pass them, there are other alternatives depending on what they know of the court - calling on contacts with other members of the royal family or their personal friends,influential nobles, information from servants, ambassadors, merchants, etc. I run long campaigns and if I mean the royal court to be important in play I will arrange early adventures so the PCs have allies who owe them favours which will by useful at this point. Maybe the king is ultimately unwilling to listen to them, but the crown prince is and offers them aid. Later in the campaign the king dies, and the PCs have already befriended the prince and are able to help him gain his crown in the succession war that follows.
If it's not important to the campaign, I won't put as much work into it and may run it as a skill challenge or simply narrate what happens if its ultimately unimportant. The more players are invested in the issue, the more attention I will pay to it.
At high levels they may already be known to the king, they may be nobles, and they are sufficiently powerful and famous that they will be given special treatment by the functionaries of the court and can expect to see the king in short order unless something is up. Epic level PCs can do whatever the hell they want at the mundane kingdom level. The particulars of different settings will modify this, of course.