Amber is the essence of longing, even when experienced. It is the perfect moment of sunrise, an eternity that you must take leave of to face the rest of the day. In short, my experience is that Amber works much better in a story as somewhere you long for and occasionally joyfully visit than it does as a permanent setting.
So I've loaded your characters up with reasons to avoid the city they were born to love and rule. Primary among those is unspoken Family tradition. Random does not spend more than two months of each year in the city, and it is expected (by the Elders) that no other Family member of adult age will do so. The explanations for this range from the cynical (any Prince left in the city long enough will conspire to sieze the throne) to the idealistic (only by removing the Family as a constant arbiter and judge can Random encourage the people of Amber toward self-determination). Your character is welcome to disagree with the policy... many do. But disobeying it would be risky.