But he was able to consolidate his power because he had the claim. If he didn't have any claim at all, other houses (especially those with Targaryen blood) could have said they were the rightful heir. It's a big part of why he is able to again unite the Kingdoms.

As for why Stannis doesn't have a claim... Because Robert's sons exist. If Joffrey and Tommen died before he took up against the crown, he would have been the "rightful heir" to the throne.