His running away could also have an impact on the village. If he was an important part of the village because of his family, his departure might cause unrest or a power vacuum. People might start to question the stability of his family or the village leadership. And if he gets into trouble while away, it could also bring danger back to the village in some way.
He could become stronger on his own. Away from his family, he'll have to rely solely on himself. This self - reliance might push him to train harder and unlock even more of his 'godlike' powers. For instance, he might find new mentors or training methods that his family would not have allowed.
One consequence could be that his friends worry about him a great deal. They might go on a search for him, which could put them in danger as well. After all, they care about Naruto deeply and won't just let him disappear without trying to find him.
He might miss out on some crucial training. Jiraiya has a lot of knowledge and skills to pass on, and by running away, Naruto could fall behind in his development as a ninja.
Maybe he feels misunderstood by his family. In many Naruto stories, his family situation can be complex. If he's 'godlike', he might have powers that his family doesn't understand or fears, leading him to run away to avoid conflict or being treated differently.
He might face a lot of danger on his own. Since he's running away from Kushina who would usually protect him, he could encounter stronger enemies that he's not fully prepared to face.
One possible reason could be fear. Minato is a very powerful shinobi, and Naruto might be scared of him for some reason. Maybe he has heard stories about Minato that made him afraid.
Naruto could end up getting lost, both literally and metaphorically. He might lose his way in the shinobi world, not having Minato's guidance. He could make wrong decisions about his training, alliances, etc.
One possible reason could be that he feels misunderstood. In the Naruto series, Naruto often faced isolation and judgment from the villagers. If this extended to his family in the fanfiction, he might run away to find a place where he is accepted. Maybe he discovers some dark secrets about his family that make him feel unsafe or betrayed.
Maybe he's running away from the pressure in the village. There are always high expectations on him, and he might feel suffocated and need some time alone to figure things out.
One possible reason is she's had an argument with her friends and feels she needs to get away to cool off.
One possible plot could be that Naruto feels constantly overshadowed by his family's expectations. Maybe his family is full of powerful shinobi and they expect him to be just as great immediately. He runs away to find his own path and prove himself on his own terms. Along the way, he meets new friends who are also outcasts and together they form a sort of 'family' that accepts him for who he is.