Rey accepting herself for who she is instead of just pretending to be Luke’s long lost daughter would be a logical end to the character arc instead of just nonsense.
I think you’re misinterpreting it. Rey’s connection to the Skywalkers comes from Leia, listen to what Luke says when he gives her Leia’s saber. It isn’t that she looks at them as her direct relation, it’s about carrying on what they started.
She see’s the Skywalker name as a name for hope.