The fact is, we are still not sure if those deaths were caused by the blood ritual.
Unlike Renly's death, where there was a supernatural being directly involved in the death (the shadow baby).
Robb's death was due to the machinations of Tywin Lannister, and executed at the hand of Walder Frey.
Nothing supernatural here that we can see.
Book spoilers:
Joffrey was poisoned, possibly by the Tyrells.
Again, nothing supernatural.
Balon Greyjoy on the other hand
we have no first hand account of his death.
But its timing coinciding with the return of Euron has led many to speculate that it was an assassination ordered by Euron.
It could simply be a lucky guess on Melisandre's side, or a self fulfilling a prophecy.
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Melisandre can kill anyone just by drawing leech
Says who?
Leeched blood is a small sacrifice, and would only yield a small favour from the red god/blood magic.
Generally with blood magic, you need a "fair trade", i.e. sacrifice a life to give life to another (as with the birth of Daenerys' dragons), or if you "cheat" the red god of a death (Arya saving Jaqen & co.), it should be repaid with another life so that you "return balance".
Mellisandre is a cunning lady though.
She gets visions of the future from the flames.
So if she knows that someone is going to die, she can make it seem like it was her doing.
This way she seems more powerful, furthering her agenda.
Or so the theory goes.
GRRM doesn't give much explanation about the nature of magic and gods.
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Melisandre believes that Stannis is Azor Ahai .
There is a prophecy that says that Azor Ahai will be The Lord of Light's champion, and fight the Great Other in a large upcoming battle.
There will come a day after a long summer when the stars bleed and the cold breath of darkness falls heavy on the world.
In this dread hour a warrior shall draw from the fire a burning sword.
And that sword shall be Lightbringer, the Red Sword of Heroes, and he who clasps it shall be
Azor Ahai come again, and the darkness shall flee before him.
It is said that Azor Ahai will "wake dragons from stone", which is doubtless the prophecy that Melisandre is trying to coerce into existence.
She has already tried to coerce Lightbringer into existence, but we know (as Maester Aemon also noted) that it is a glamour, not a true burning sword.
We also know that Daenerys has already awakened her dragons from the stone eggs, thus fulfilling the prophecy.
As for the bloodfilled leeches...
We know for a fact that Melisandre can see the future in her fires, and that this is something she is very good at.
So what most likely happened is that she saw that Robb, Joffrey and Balon would die, so she decided to put on a show for Stannis, to convince him that he needed to give her the boy to work her magic.
To convince him of the power of King's Blood.
She believed that if she did indeed sacrifice the boy, a dragon would awaken from stone, and the prophecy would be fulfilled.
From a more pragmatic point of view, a dragon is also a sign of power, and it would be easier to get support from the lords of Westeros with a dragon.
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I thought he was the illegitmate son of Robert Baratheon.
In her magic she uses there is 'power in a king's blood' and hence she uses his blood to power her magic.
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We don't necessarily know that she does want to kill (Edric Storm Book/ Gendry TV), just that Davos is worried that she is going to kill Edric/Gendry and so takes him to safety.
We haven't actually seen her kill anyone (other than Renly)
just that her mutterings about the power of King's blood makes everyone else think she is planning to sacrifice people.
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Melisandre can just be seeing into the future where see catches glimpses of what is to come and then use that to "convince" others that by sacrificing Edric and Gendry she could do much more.when in fact she just wants to REMOVE any threat the her Kings rule.
Both Edric and Gendry could in theory lead an uprising and overthrow Stanis as king.
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Melisandre names 3 names as she performs her blood ritual.
Melisandre: The usurper Robb Stark.
The usurper Balon Greyjoy.
The usurper Joffrey Baratheon.
Game of Thrones, Season 3
Episode 8
, "Second Sons" Lets look at how each of those died.
Robb Stark Died in the Red Wedding as planned by House Frey, House Bolton and House Lannister.
The Frey's did it to gain power with the Lannister.s and to take revenge on Robb betraying them by marrying Talisa instead of a Frey girl.
The Bolton's because they've had a long history of fighting with the Stark's, to take power in the North by claiming Winterfell and to gain power with the Lannister's.
The Lannister's did it because they were at war with the Stark's and it gave a nice neat ending for them.
Balon Greyjoy Was killed by his brother Euron Greyjoy who did it in an attempt to take power of the Iron Islands.
It was a risky move as the next king is chosen by a Kingsmoot and Yara
was favourite, especially as Euron had been AWOL for a while.
Joffrey Baratheon Killed at the Purple Wedding by Petyr Baelish and the Tyrell's by poisoning his wine.
Petyr did it to cause chaos so he could climb the ladder.
Also so he could sneak Sansa Stark away with him in the following chaos.
The Tyrell's did it as Ollena did not want Margaery to marry Joffery.
As you can see as each died because of someone else, that was already seemingly planning to kill them
it's unclear if there deaths were due to the ritual.
As for why she wants to kill him, well more of a sacrifice equals a more powerful outcome.
A great gift requires a great sacrifice.
Game of Thrones, Season 3
Episode 10, "Mhysa"
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If Melisandre can kill anyone just by leeching Gendry, why kill him rather than keeping him alive?  If he is alive he can be leeched at will and allow her to kill as many people as she likes.
Why does Melisandre want to sacrifice Gendry?
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Melisandre may just be seeing into the future where catches glimpses of what is to come and then uses that to "convince" others that she could do much more by sacrificing Edric and Gendry. However, she just wants to REMOVE any threat to her King's rule.  In theory, both Edric and Gendry could lead an uprising and overthrow Stanis as king. Melisandre knows she is capable of so much more if she sacrifices Gendry and the threat of his rule. 
Melisandre knows she is capable of so much more if she sacrifices Gendry and the threat of his rule. 