It was because of Kurama's Yin Chakra inside him. If Minato was suitable to become Kurama's Jin when he was alive then Konoha wouldn't have needed to import Kushina from Uzushiogakure to become Kurama's host. Minato is in Edo form so he is able to handle it.
Sorry I think you're overlooking the obvious. Why did Kishi unveil who Naruto's mother's ancestry was, but not his father? Oh yes because he had limited chakra.. no that was just an excuse to not do it. He didn't do it because who his father lienage is the direct branch to the SO6P. This is further backed up by exactly 2 people have been shown to be able to enter that particular chakra mode.
So what's special about the sage's bloodline then that lets Minato and Naruto enter that particular chakra mode? I believe it's the ability to balance & harmonize their own chakras with other chakra. Essentially their greatest ability is to strengthen others.
We saw this already when Naruto fused his Chakra with the 9 tails and the Shinobi alliance.