.....Shiro turning into a Robeast, worse than dying.
Shiro making Keith promise to kill him if he transforms.
Keith refusing, vowing to find a way to save him.
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.....Shiro turning into a Robeast, worse than dying.
Shiro making Keith promise to kill him if he transforms.
Keith refusing, vowing to find a way to save him.

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Theory: Shiro doesn’t die, but...
...Keith still becomes leader of Voltron.
First of all, going by the tone of the show, it seems unlikely to me that any of the paladins will die. Going by Avatar and Korra, it seems unlikely to me that any central character will die. That doesn’t mean that they’ll all be just fine, though.
So I think that Shiro will survive, but be forced to step down as leader. Possibly even stop being a paladin. It does seem like Keith is ‘destined’ to become leader, and at the moment, I can’t say that anyone’s positions are set in stone.
There are a few reasons I’ve thought of for this theory:
Shiro’s existing disability. He loses his prosthetic arm, and there is a considerable amount of time before a new one can be made and fitted. During this time, he can still fly, but needs to swap positions with another paladin--either with Keith directly, or everyone ‘reshuffles’ and ends up with a different lion. By the time Shiro can pilot Black again, Keith has come into his own, and continues to lead.
Another injury. Shiro is physically out of action, possibly comatose, and he needs extensive time in the pods, or the pods are not available for a while. Allura steps up as a the blue paladin, Lance pilots Red, and Keith becomes leader in Black.
Mental trauma (the most likely one imo). Shiro’s ptsd gets worse, to the point he cannot lead. Maybe he can’t even pilot, he gets flashbacks just stepping into the lion, and Allura becomes a paladin as in theory 2. Shiro is still a big part of the series; he can still guide his team and provide backup alongside Coran. However, he will never pilot Black again, except apart from an episode in a later season where the situation forces him to--and that would signal the start of his recovery.
So, that concludes my rather long way of saying ‘Shiro will (probably) live, but he may not be OK’. Any thoughts?