There are roughly 5/6 parts to my game.
As i said i will not refute your answer, but just go onto the next point.
First point is the shiki fujin.
To make life easier ill bring links.
Hiruzen uses the shiki fujin.
As we know the demon's hands enters the user's body and uses the body as a medium.
When hiruzen pulled the 2 hokage's souls out this is what happened.
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Some might say he sealed their souls in his body.
But we all know that it isn't true because a few pages later, after getting orochimaru's soul we also see the hokage's souls (faces), along with oro's and hiruzen's attacked to the death gods
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He then takes out his knife and cuts out the souls and eats them
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Prior to the link above hiruzen said
"With this jutsu the one whose soul is sealed will suffer for all eternity in the belly of death"
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So hiruzen at first appeared to have sealed the hokage's in himself, but rather he was merely a medium and the hokage's were actually tagged to his soul which is also tagged to the death god. When it took its knife out, he eat their souls.
Same thing happened with minato
Majority think this scan shows that he sealed kurama in himself, but actually its the same thing hiruzen did at first.
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Few chapters later we saw his the portion of kurama attached to minato's soul which is attached to the death god
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On that same page you'd note that the death god has taken its knife from his mouth and is about severe the link and devour both minato's and that portion of kurama.
To put it simple the exact same thing that happened with hiruzen happened with minato.
That portion of kurama along with minato's soul was devoured by death god. He didn't or should i say was not supposed to die a jinchrucki.
Next question do you why, minato disagreed with kushina sealing the fox and dying with the fox?
Dude i said ill ask the questions and bogard answers, but since you are taking his place you do the same
To answer your question its not that minato cannot become a jin, but rather in this scenario it brings out inconsistencies