The scenes between Arya and Tywin are awesome, but they also
show hints of the problems that would eventually ruin the show,
because they don't really hold up logically. Who does Tywin think Arya is?
He knows she's not the commoner she's pretending to be, he knows she's
from the north, who he's fighting a war with. He believes Arya Stark is
safely being held captive in King's Landing, so he wouldn't think that was
her, but she is at minimum a valuable hostage who could be held for ransom
or used a bargaining chip and at worst a potential spy in his midst, listening
in on his war councils and reading his secret messages. Especially when
people started mysteriously dying around him, Arya should've been among the
first he had Put to the Question to find out who she was and what she knew.
You can say that Tywin liked Arya. That he respected her intelligence, her
cunning, that she reminded him of himself or of a young Cersei, and that's
fair. But one thing that has never been said about Tywin Lannister is that
he is someone who lets sentimentality get in the way of advancing his
cause. However much he might've liked Arya, he wouldn't have let this
northern noble just hang out in his war room when he had other better
uses for her.
D&D have shown they are good writing scenes like this. Seasons
7 and 8 are full of scenes that look awesome in a vacuum,
that give their characters plenty of opportunities to look
and act cool, but don't stand up to logical scrutiny,
and this I think is an early example.