What was the point of Stannis in the show?

Stannis is done. I wouldn't expect anything else in the show to progress his character.

He's still alive in the books tho as far as we know. Unless we are trusting a letter written by Ramsay Bolton.

As also-rans go he had a good run. I actually thought for a bit he might end up taking Westeros.

Never liked him though - miserable self-righteous git who burnt his daughter and murdered lovely Renly.

It's hard not to like Mel also.

Sure she's done horrible things, most characters have. And I still don't think Rhllor is a good God to follow even if he is the strongest. But the way she carries herself and her theme music just makes me love her scenes so much. Her lines tend to sound almost poetic.

Shireen was sacrificed so the battle would be easier to shoot and more spectacular, you ingrates 

What is the purpose of a character in a story? I guess it depends on the writer. In the case of GRRM, it seems characters are their own purpose beyond what they achieve or introduce in the overall plot. Yet Stannis was first presented as the allegedly rightful but undesirable alternative to the new king.
The conflict between the rightful and the desirable is at the heart of the story and so the purpose of Stannis was to present readers or viewers with a difficult choice. 
Stannis was meant to make you root for the Lannisters at the battle of the Blackwater. You were supposed to question the whole pro-Stark, "let's just kill Joffrey and go home" nonsense by realising this whole bloody war is a deadly mess no one should actually want. He was meant to make you question whether Joffrey really was that bad in the end. Yes, a boy murdering prostitutes for fun is an abomination but so is a king who burns unbelievers at the stake.

The show botched the ending, but be certain the books will not have him achieve more even though his demise will make more sense.

Renly left with Loras and further knights the day Robert died, before any war was started. Ned Stark's intention was to stage a swift coup to establish what he considered the unique, rightful order. Renly openly intended to do away with these notions and offer himself has "a better pretender" as opposed to the rightful one. Since anyone can decide he makes a "better pretender", he was opening competition and changing the game, moving from following rules to "let the stronger win".

Ned, Robb and Stannis would all have agreed to see Stannis on the throne but Renly would dispute it. "The men carrying these bolts of cloth will make me king", he said, not any notion of right or wrong other could agree to.

Robert had set no precedent. The Mad King had proven unworthy of the allegiance he demanded by murdering his vassals. Robert's rebellion was not a war for the throne, it was the justified taking down of a threat.

Stannis could and most likely would have gone to war if Ned had put Renly on the throne. His "claim" was a time bomb. Might only makes right after the fight is settled. Right, however, gives one better chances of rallying the winning side because the first who declare themselves take a bet on what will follow. They are more likely to declare for those they think have the right on their side, then it can snowball. In that way, right makes might to start with.

Renly could overcome that phenomenon because the Reach and Stormlands together commanded such a power but elsewhere, the common opinion was that he wasn't the candidate to support. We saw Robb discuss this with his bannermen just before they made him king. He was not willing to support Renly's claim.

A rebellion is not a succession. While the latter can rely on rules and agreements to avoid violence, a rebellion is resolved with the damage already done or at least ongoing and mechanisms are different.
The Targaryen heirs were murdered to stabilise the new power. The rebellion had established a new balance of forces in which the "legal" successors of the overthrown king had no rights. They would not inherit from a violently deposed king because their natural allies would have been those who stood against the winning side. There could have been a better settlement. Robert could have married a Targaryen daughter to merge allegiances (this is what Henry VII did to end the Wars of the Roses) but I'm afraid there wasn't any and Tywin Lannister had other plans anyway.

Renly would have been part of the Baratheon dynasty, but he would have had to eliminate Stannis in a definitive manner.

Looking back at England, I cannot help seeing some sort of parallel with William the Conqueror whose dynasty was all but stable in its first century of existence. It remained on the throne but brothers and cousins fought each other until Henry II found the sheer strength to hit hard on every head that popped up unasked and had the political sense to delegate power to trustworthy vassals (something maybe akin to the Wardens Aegon named). 
I might be going out on a limb there but it feels like the rupture of the Anglo-Saxon "right" William brought with him was a significant part of the causes for unrest. The old order which had kept the Anglo-Saxon dynasty going smoothly (unless when overthrown by Danes) was no longer present within the new Franco-Norman situation. The change of paradigm (and perception of right) might well have been the cause for what was to be quite a rough ride for nearly a century.

William's successor William II and his Tyrell (boy)friend were eliminated roughly, by the way. Is it a coincidence?
William II also had to face rebellion from his elder brother Robert, Duke of Normandy, who might have resented not being given England by his father…Did that bloody GRRM steal everything?

GRRM describes King Stannis as seasoned battle commander. He is the only person described as giving chills to Lord Tywin Lannister. Renly was no king. He was a sweet faced boy who got some support because of his lover Ser Loras. Real traditionalists/conservatives such as Lord Stark knew that he was the claimant of iron throne based on inheritance rights. Of course the option to steal the iron throne was always open to everyone including to Renly.