This is my theory as to why Minato only sealed away the Yang chakra inside of Naruto, also why and who summoned the Kyuubi in the 1st place. And if u think about it will make sence...

The Kyuubi (9 tailed fox) is an ageless creature souly made up of immense chakra. A long time ago the Kyuubi formed a pact with one Uchiha Madra who was one of the two founders of Konoha (along with his best friend Shodaime Hokage) as well as the creator of the Uchiha Clan, Madara was the son of the "Ten Gu King" a full blooded demon. Madara merged with the Kyuubi in order to defeat his father (who wanted his sons body because of its power; the sharingan).

After somehow summoning the Kyuubi to this world Madara merged with the Kyuubi and became the 1st ever jinchuuriki, along with the Kyuubis power Madara was able to seal away his father the "Ten Gu King". After doing so Madara knew that the only was to unseal his father would be with three Mangekyou Sharingan users and the Kyuubis chakra. Now since the Kyuubi is ageless creature Madara knew that at some point it could be captured.

Thats when Madara decided to keep half of the Kyuubis power inside of himself and let the other half go so that it would be much harder to unseal his father. But what Madara didn’t know was that the half he kept for himself was the Yin (bad or evil) chakra of the Kyuubi which eventually corrupted Madar and led him to attack the very village he helped create and enter a battle to the death with his best friend Shodaime Hokage.

When the Kyuubi attacked Konoha 14 or 15 years ago Minato sealed the Yang (good) part of the Kyuubi inside of his son Naruto because there was no Yin chakra to be sealed. This helps explain y the Kyuubi had attack Konoha in the 1st place, it was because the Kyuubi was searching for its other half the half that Madara had. (Madara was from Konoha) So it makes sense y the Kyuubi would come back to look for its other half.

The thing the Minato knew that was of “grave importance” as Jiraiya mentions in chapter 370 was the probably the fact that the Kyuubi attacked Konoha because it was searching for its other half and thats y there was no point in killing the Kyuubi(The Reaper Death Seal is designed to kill) so Minato allowed Naruto access to the Kyuubis Yang chakra so that Naruto would have some power to protect himself and the ones around him when eventually enemies would come searching for the Kyuubi (Akatsuki).

Also that is y Jiraiya spent 2/12 years trying to teach Naruto how to control the Kyuubis chakra so that when Naruto would use the Kyuubis chakra it wouldn’t take him over. Because there is no other reason for Jiraiya to teach naruto how to control the Kyuubis chakra. But as we have seen Naruto isent ready for that much power and totally loses control when he uses 4 tails. I also think that "The Jutsu" Jiraiya and the frog referd to has something to do with Narutos "Ultimate Rasengan Jutsu".

***Note: I seriously doubt that the Kyuubi would just randomly attack Konoha unless it was searching or wanted something (its Yin half) which i think Madara had or has in him or hid somewhere in Konoha and since Madara was from Konoha it would make sense for the Kyuubi to look for its other half there.***