As with many things it could have been quite different. This was all part of Martin’s initial scheme, which would obviously change as he began to write and build his world, including Tyrion in Game of Thrones.
Although Game of Thrones’ Tyrion is no stranger to murder and other heinous crimes, committing such an act against the main heroic family would have painted him quite differently.
It’s true that the shades of gray and the complexity of its characters is one of the reasons Game of Thrones turned out to be so compelling, but this probably would have turned viewers against Tyrion, at least for a while.
With him burning Winterfell, Catelyn Stark along with Bran and Arya, would flee to the Wall. That would mean that Tyrion had been involved in the deaths of Robb and Catelyn, as well as the destruction of Winterfell in Game of Thrones.
These are events that would make it difficult, though perhaps not impossible, for him to become the beloved protagonist that he is. There was some hope for redemption in Tyrion’s original story, as he would kill Joffrey and be exiled for it on Game of Thrones.
Trying to find some peace with the Starks. But things would get even more complicated with the emergence of a new romance, as Tyrion realized that he was in love with Arya, who, in turn, loved Jon Snow on Game of Thrones.