Starfleet doesn't have the death penalty, except for General Order 7. And that has nothing to do with a mutiny.
Also, how often did Kirk ignore the Admiralty? Insubordination? Apparently not as bad as mutiny.
If Kirk committed mutiny, this discussion would be how unfair Starfleet was to him.
Some have interpreted the line as inferring that Spock would pay for it with his life.
T'Pau says "Live long and Prosper, Spock"
Spock says "I shall do neither, for I have killed my captain, and my friend."
I don't see a reason for Burnham to be made Captain. I do see her having to take over command of the Discovery at some point, in a crisis, but not permanently.
She was stripped of rank. She's no longer an officer. And whether or not it was deserved, she was sentenced to life in prison. Let's say she saves earth(or maybe Antos IV) and the President of the Federation pardons her, or whoever has that power. Do they give her her rank back? And then some?
If Spock really had killed Kirk, even though it was in a sanctioned duel, would he still be in Starfleet? He didn't seem to think so.
Also, she's the main character of a show that was promoted as specifically not focusing on the Captain. And she wasn't cast to play a captain. The only way I see it happening is in the final episode to feature Burnham showing her in command of her own ship in the closing shot, and showing her journey complete, from that first conversation when we were introduced to Burnham. A story concluded.
Other than that, I have a very difficult time picturing "The adventures of Captain Burnham."