In some regards, this is a useful way to plot a game. The characters have a clear and fairly well defined goal, even if there isn't a clear way to go about doing it. They can work together or seperately as fits their style. It's also familiar. Most RPG campaigns are built around similar lines.
On the other hand it does have some downsides. If the players are clever, lucky, skillful, or all of the above, they can disrupt the whole campaign by winning too soon, upsetting the plans of the Forces of Evil beyond repair. Or they can fail to take the bait, deciding they're uninterested in stopping what might seem to be a minor threat. Oops, there goes the universe. It can also have problems with logistics. If the threat is really that large, characters and other memebers of the Amber royal family will put aside their differences to help destroy it. That's fine, but it can make the game a bit boring, especially if the family has grown and there're enough people that when banded together they become an unstoppable force.
Either way, it's not the only way to run an Amber game. Amber, more than many other genres and settings, has many ways to play without having a unified threat. In fact, it can be run without an external threat at all. Here's some ideas for doing so:
This is perhaps a defining quality for Amber. The entirety of the first several books is devoted to what amounts to a war of personal vendetta between Eric and Corwin, and fueled by aid of others with their own scores to settle. Forming tenuous alliances, backstabbing, betrayal, and switching sides in mid-intrigue are all seemingly expected behaviors for the Elders.
It's not hard to extrapolate such a scenario out into a full campaign. Set up a relationship map or flow chart of all the possible reactions and interactions between the Elders, get the players to add in their own alliances with each other and the elders, and turn the whole thing loose to see what happens. The end result could be endless sniping and uneasy tensions all the way to formenting a full out war in the style of the original Chronicles.
This style of play would probably work best with a group that really enjoys just 'being' their character, spending time doing non-plot interactions like just having dinner with the family.
Amber isn't necessarily the only game in town. Their relationship with the Golden Circle is important enough that they need to keep civil relations with those country-shadows, lest they find themselves fenced in on all sides by united enemies. Similarly, in an Amber game where the PCs are 4th or 5th generation removed from Oberon, the Royal family might be large enough that it has it's own internal politics much like the Great Houses of Chaos.
In a world where physical peace is enforced by either social convention or even impartial bystanders, players might have to beg, borrow, trade favors, and forge alliances to promote their own goals. Imprompteau coalitions, strange bedfellows, and intrigue are sure to abound.
Works best if the players can divorce themselves from the characters enough for in-game backstabbing to be friendly.
This isn't strictly the same as the above two. Instead, there's a threat, but it's unknown to the majority of Amberites, and perhaps unbelievable. A shadowy consipiracy is out to do something rotten, and only the PCs know. Or maybe they don't know, but have suspicions based on a few weird encounters. This works particularly well if the characters are all brand new to the family or children of exiles or traitors, and suspected of same, so that no one believes their claims. Potentially the PCs could even end up working at cross odds with the Elders, as they've been told to back off and drop it, or are otherwise sneaking around to find out what's going on. The threat could be even come from within the family. Perhaps the redheads weren't so quick to give up their conspiracy as they claimed. Or Caine and Gerard and Julian are sick of playing second fiddles to Corwin and Eric and Bleys and have decided their time has come. Good for mystery, suspense, intrigue, and plenty of sleuthing.