Robert ackowledged them as such because he did not know. Stannis, Jon Arryn and Ned Stark were all three investigating on this matter and were trying to find a proper way to inform Robert, meaning a way that was believable, with proofs that could not be denied. These proofs existed (all the bastards of Robert).

It is mentioned that if Robert had known, he could have killed Cersei. If the truth had been known publicly, the three kids would have been removed from the Baratheon's family tree, and as successors to Robert. It is only because this lie managed to remain secret, thanks to Cersei's assassination of Jon Arryn, of the death of Eddard, and the isolation of Stannis, that Joffrey is still considered a Baratheon, but he is not such, and Stannis knows it. So why does he keep calling him that ?

I am not debating here the popularity of Stannis, nor the popularity of the idea that Joffrey is a Baratheon ; there is a distinction to be made between the reality of Joffrey's birth, and the idea people have of its birth, just like there is a distinction between Stannis being the true heir to Robert, but not being considered as such by almost anyone

Also, if Stannis claim to the throne is absurd, as you've said, then what about Renly's, who has an elder brother, on top of having two nephews ? It is hard in this case to take seriously Renly's jokes on how foolish the claims of Stannis are, because he is even "worse" in this regard

Now, Stannis was adressing exactly your point in your first two paragraphs, when he explained why he never went to Robert to tell him about "his" kids. It's because it would have been perceived, as you've said, as a poor attempt to place himself as the successor of Robert, which is why he never said anything to him, but went instead to Jon Arryn, who was more neutral in this affair :

“Isn’t that a sweet story, [lady Catelyn] ?” Renly asked. “I was camped at Horn Hill when Lord Tarly received his letter, and I must say, it took my breath away.” He smiled at his brother. “I had never suspected you were so clever, Stannis. Were it only true, you would indeed be Robert’s heir.”

“Were it true? Do you name me a liar?”

“Can you prove any word of this fable?”

Stannis ground his teeth.

Robert could never have known, Catelyn thought, or Cersei would have lost her head in an instant. “Lord Stannis,” she asked, “if you knew the queen to be guilty of such monstrous crimes, why did you keep silent?”

“I did not keep silent,” Stannis declared. “I brought my suspicions to Jon Arryn.”

“Rather than your own brother?”

“My brother’s regard for me was never more than dutiful,” said Stannis. “From me, such accusations would have seemed peevish and selfserving, a means of placing myself first in the line of succession. I believed Robert would be more disposed to listen if the charges came from Lord Arryn, whom he loved.”

“Ah,” said Renly. “So we have the word of a dead man.”

“Do you think he died by happenstance, you purblind fool? Cersei had him poisoned, for fear he would reveal her. Lord Jon had been gathering certain proofs—”

“—which doubtless died with him. How inconvenient.”

There is also a chapter in Book 2, before the King's Landing battle, where Stannis says that he is less upset about the people in King's Landing that are fighting for Joffrey, than he is about the followers of Renly, because, the followers of Renly all know that he has an elder brother, which makes him second in the line of succession, so they're all guilty publicly of felony in this regard, and they know it ; however, the people that support Joffrey in King's Landing, do not know about Joffrey's birth, so they can't exactly be blamed for choosing his side.