If the Starks are synonymous with justice and prosperity in the North, do you think they would be mythologized in northern culture if they are never reestablished in Winterfell?

They already are:

“When there was a Stark in Winterfell, a maiden girl could walk the kingsroad in her name-day gown and still go unmolested, and travelers could find fire, bread, and salt at many an inn and holdfast. But the nights are colder now, and doors are closed. There’s squids in the wolfswood, and flayed men ride the kingsroad asking after strangers.”