Why did Kurama keep helping Naruto?
The reason might be because if the Jinchuuriki dies, the bijuu will indeed be set free. However, the bijuu will also "die" first. The bijuu will "die" first before reviving itself after some time. Where it revives might also be random, as seen in Isobu's case. Isobu, or more commonly known as the Sanbi, revived itself after its host, the 4th Mizukage died. It revived itself in a lake.

Now, Kyuubi might want to avoid this. There are three possible reasons for this.


Kyuubi had half of his chakra sealed in the 4th Hokage with Shiki fuuin. Because of this seal, even if the Kyuubi had revived itself, it would have revived with only what he had during the time he was sealed in Naruto.
Kyuubi dislikes the idea of having to die first. Kurama's nature is a bit prideful, so that fits.
The half of Kyuubi that was sealed in Naruto was the less sinister half, and Naruto had the power to attract people, so this half might have taken interest in Naruto ever since Naruto used his power to defeat Haku.