In S4E8 of Game of Thrones, the Hound and Arya finally reach the Eyrie and are informed of Lady Arryn's death.
In S4E10, Brienne finds Arya and the Hound ten miles away from the Eyrie, leaving.
Why would the Hound not have left Arya at the Eyrie?
I understand that they don't realize that Sansa is inside, but even still: the hound was looking for someone to ransom Arya to, and even though Lady Arryn is dead her son is still Arya's cousin, and Littlefinger is incredibly sympathetic to the Starks due to his relationship with their mother...?
The Hound tells Brienne that "no-where is safe" for Arya, but surely the Eyrie would at least be sympathetic and allow them reprieve and shelter? Clegane must know this; his brothers cruelty towards him was a violation of the very blood-debt that every other family in Westeros displays, so he's surely aware of the obligation to shelter Arya?
I've looked on the web for answers, and so far none have been satisfactory. Even the New York Observer drops a bollock, claiming they never declared Arya's name at the gate so they left whilst they were still anonymous, which is patently untrue.
I'm reading the intended sequence of events as:
They were at the gates of a fortress, declared their name and intentions, and were told that their audience could not be granted because Lady Arryn was dead. Clegane decided the Eyrie would not be safe, so walked away.
having announced their identities, one of whom being the highest valued fugitive in Westeros; if the Eyrie was hostile, wouldn't Clegane have expected them to chase him, or arrest him? doesn't the absence of this reaction indicate sympathy, neutrality or at least indifference? where was the threat?
Was there a real, solid reason why Sandor Clegane wouldn't enter the Eyrie, or is this just to serve the plot and provide a pretext for them to become transient again?