Because once Joffrey took the throne, even with a regent, it would have legitimised him - and even if Ned remained regent (questionable that he would have even without confronting Cersei, as the Lannisters would have wanted their own as regent), Joffrey would have ended up with enough support that it would take a civil war to depose him. Ned was trying to preempt all that - plus, his honour drove him to prevent a bastard from outside of the Baratheon line from taking the crown.
The crowning of Joffrey would have been a problem for Littlefinger and Renly, too. Particularly Renly, who also had to contend with the fact that he had to install himself before Stannis arrives to make the stronger claim. And it is not clear that Renly had no idea of Joffrey's lineage - I think it is almost certain that Littlefinger was conspiring with Renly, as well as many others.
I also think you're mistaken in "Littlefinger following Ned". It is very clear that Littlefinger is playing many sides against each other through the books - and he is, in fact, almost urging Ned Stark into action.
While Littlefinger doesn't have much to lose by having Joffrey on the throne, he is constantly seeking to improve his position. He is hoping to gain from whoever is seated the throne - it seems that he is trying to get Ned to work with Renly. When Ned fails, Littlefinger simply turns back to the Lannisters and betrays Ned, as he sees that as his best opportunity for advancement.
There actually is more than one motivation here - Renly may not be aware of Joffrey's true ancestry. Just before Roberts, he is trying to prevent the Lannisters from gaining the regency and hoping to get Ned to support him in this - whether Joffrey is on the throne or not is immaterial to this. As long as Joffrey is underage, the regent will be the "true" ruler of the kingdoms (as we see later on, the real powers are not with the regent, but with people like the Boltons and Varys and Tywin Lannister).
However, when Renly can't get Ned to support his plan, he leaves King's Landing. When Robb and Stannis declare war on Joffrey and the Lannisters and declare their own crowns, Renly decides to pitch in his own claim for the crown.