Renly Baratheon pragmatically summed it up in A Clash of Kings (p. 265/761 of the 1999 Bantam paperback):
Tell me, what right did my brother Robert ever have to the Iron Throne?
Oh, there was talk of blood ties between Baratheon and Targaryen, of weddings a hundred years past, of second sons and elder daughters.
No one but the maesters care about any of it.
Robert won the throne with his warhammer.
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Robert had a Targaryen grandmother, and was the closest person to the legitimate line of succession who wasn't a descendant of Mad King Aerys.
So he had a much better claim than Tywin.
See the Targaryen family tree .
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Tywin wasn't part of the rebellion, at first - he was the Hand of the Mad King till he switched sides.
Robert led the rebellion against the Targaryens, and by that virtue seized the Iron Throne.
I'm sure the fact that he also had the largest army, and his brother Stannis
the fleet assured him the throne.
I don't think the act of killing the heir was the reason - the king was killed after the heir apparent, by a Lannister
, you'd think that would influence who gained the throne.
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"Tell me,
what right did my brother Robert ever have to the Iron Throne?
" He did not wait for an answer.
"Oh, there was talk of the blood ties between Baratheon and Targaryen, of weddings a hundred years past, of second sons and elder daughters.
No one but the maesters care about any of it.
Robert won the throne with his warhammer."
A Clash of Kings - Catelyn II
This quote is always the starting point when talking about how Robert came into the crown, and I think we all know the general overlay of Robert's Rebellion well enough.
The answer I believe is both , but also I don't think it could have happened any other way.
Remember Robert did not start the rebellion!
Sure he was upset at the "abduction" of his promised Lyanna, but it was Rickard and Brandon Stark who rode to King's Landing to demand justice and Jon Arryn who first called the banners.
Instead of granting them fair hearing, King Aerys had them brutally slain, then followed these murders by demanding that Lord Jon Arryn execute his former wards, Robert Baratheon and Eddard Stark.
Many now agree that the true start of Robert's Rebellion began with Lord Arryn's refusal and his courageous calling of his banners in the defense of justice.
The World of Ice and Fire - The Fall of the Dragons:
Robert’s Rebellion
We know the ending...
the rebels won and Robert was crowned king, even if did not quite want it himself.
Robert sat down again. "
Damn you, Ned Stark.
You and Jon Arryn, I loved you both.
What have you done to me?
You were the one should have been king, you or Jon.
" "You had the better claim, Your Grace." A Game of Thrones - Eddard VII
But why did Robert have a better claim?
To answer this let us think like a Maester and ask what the Iron Throne represents...
It represents the Targaryen rule of a united Seven Kingdoms.
So the ruler sitting on the Iron Throne should be a Targaryen.
Well, all the of true Targaryens were dead or in exile, and of the victors who had at least some Targaryen blood in their veins...
it wasn't Tywin Lannister , it wasn't Jon Arryn , and it wasn't Eddard Stark .
It was of course Robert Baratheon .
Robert's lineage didn't even need to go back that far, only to his grandmother Rhaelle Targaryen , to find it.
As they say history is written by the victors and since Robert was the ultimate victor the stories and songs will focus on his exploits.
Robert was instrumental on the battle field but he also had the right blood in his veins to ensure a decisive and complete tranisiton of the Iron Throne to a non-Targaryen family.
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Robert won the throne.
He killed Rhaegar Targaryen on the Ruby Ford and Jaime Lannister killed King Aerys in the throne room in King's Landing.
Eddard Stark sacked King's Landing.
He went into the throne room still on his horse and Jaime was sitting there on the Iron Throne.
Jaime yielded to Eddard Stark's icy stare.
Tywin Lannister was allowed to come into the gates of the city when Varys tricked Aerys into believing he was there to help him.
He wasn't.
Tywin gave the order for the murder of Rhaegars children and Elia Martell.
Later, Robert and Eddard spoke: A Game of Thrones - Eddard VII
Robert sat down again. "
Damn you, Ned Stark.
You and Jon Arryn, I loved you both.
What have you done to me?
You were the one should have
been king, you or Jon.
" "You had the better claim, Your Grace.
" "I told you to drink, not to argue.
You made me king, you could at least have the courtesy to listen when I talk,
damn you.
Look at me, Ned.
Look at what kinging has done to me.
Gods, too fat for my armor, how did it ever come to this?"
So, Ned could have taken the throne but did not want it.
Robert didn't really want it either, he wanted Lyanna.
He did have a better claim than Eddard but even so, many called him a usurper.
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How did the crown come to land on  Robert's  head versus  Tywin's , say?  Was it because Robert killed the heir apparent and/or had command of a sizable army?
Did Robert Baratheon earn the crown by his exploits or was it his right by succession?
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scifi.stackexchange.com/users/584/Nick T
Robert won the throne with his warhammer. He  led the rebellion against the Targaryens, and by that virtue seized the Iron Throne. Robert had won the throne by leading the rebellion against the Targaryens. So in the end "Robert won the throne with his warhammer".
Robert had won the throne by leading the rebellion against the Targaryens. So in the end "Robert won the throne with his warhammer".