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.
One possible reason could be that Naruto might be afraid of the intense training Jiraiya has in store for him. Jiraiya is known for his strict and grueling training methods, and Naruto, being young and sometimes impatient, might just want to escape that for a while.
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 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.
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.
Perhaps Naruto feels that Kushina is being overprotective. He wants to prove his independence and go on his own adventures without her constant supervision, so he decides to run away.
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.
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.
One consequence could be that Sakura might feel neglected or confused. If Naruto has always been a part of her life and suddenly stays away, it would definitely have an impact on her emotions. She might start to question what she did wrong or if Naruto has changed in some way.
Another reason could be for the sake of protecting the village. Maybe he discovers some sort of threat that he believes he can only face by leaving Konoha. He doesn't want the village to be in danger because of him, so he runs away to draw the danger away from Konoha.